Build a Geometric Art Gallery in Desmos

Build 4 interactive Desmos Geometry constructions proving key theorems.

Introduction

30 Second Summary

Geometry is inherently visual, but textbooks flatten it into static diagrams that never move. Imagine being able to drag a point and watch an entire theorem prove itself in front of your eyes.

In this project, you will build four interactive constructions in Desmos Geometry that visually demonstrate theorems your class has not covered yet. Each construction becomes a shareable piece in your Geometric Art Gallery.

What You'll Build

A collection of four dynamic Desmos Geometry constructions, each teaching a major geometry topic while producing something visually impressive and interactive.

By the end of this project, you'll have:

A Parallel Lines and Transversals Explorer that reveals all 8 angle relationships updating live as you drag.
A Triangle Centers construction plotting the centroid, circumcenter, and orthocenter to reveal the hidden Euler line.
A Transformation Art piece built from a single polygon using reflections, rotations, and dilations into a symmetrical design.
An Inscribed Angle Theorem Explorer proving the inscribed angle is always half the central angle as you drag points around a circle.
Secret Mission: Create an animated spiral pattern using composite transformations and a slider that plays automatically.

Are there any prerequisites?

You need basic knowledge of points, lines, and angles. Familiarity with Desmos is helpful (the graphing calculator version is fine). No sign-up or installation is required to start building.

Get Oriented in Desmos Geometry

Desmos Geometry is a separate tool from the Desmos graphing calculator you may already know. It gives you a construction-focused workspace where you build with points, lines, and shapes directly on a canvas.

In this step, you'll explore the interface and practice the core actions you'll use throughout the project: placing points, drawing segments, and finding the transformation tools.

In this step, get ready to:

Identify the four main areas of the Desmos Geometry interface.
Practice placing points, drawing segments, and undoing actions.
Locate the Transform and Construct menu and the expression list geometry functions.

Identify the four main interface areas

Before you start building constructions, take a moment to identify the four areas of the workspace you'll use throughout this project.

In your desmos.com/geometry tab, look at the top of the screen to find the geometry toolbar. This is where you select tools like Point, Line, and Circle.
Look at the large central area. This is the graph paper where all your constructions appear.
Find the expression list on the left panel. This is where you'll type functions and see your construction elements listed.
Just above the expression list, find the token navigator. This helps you reference specific objects in your expressions.

What are these four areas for?

The geometry toolbar is for drawing. The graph paper is your canvas. The expression list is for typing precise functions and definitions.

The token navigator lets you insert references to objects you've already drawn. You'll use all four areas as you build your gallery pieces.

Practice with the toolbar

Every construction starts with placing points and connecting them. Let's practice the core drawing tools and then find the transformation options.

Select the Point tool from the geometry toolbar at the top.
Click three different spots on the graph paper to place 3 points.
Switch to the Segment tool in the toolbar.
Click your first point, then your second point to draw a segment between them.
Repeat to connect all three points into a triangle (three segments total).

You should see a triangle made of three segments on your canvas.

Click on your triangle to select it.
Open the Transform and Construct menu in the toolbar.
Browse the available options: Translate, Reflect, Rotate, and Dilate. You'll use all four of these across the project.
Close the menu without selecting anything.
Undo everything by pressing Ctrl+Z repeatedly (or Cmd+Z on Mac) until the canvas is blank again.

Why undo everything?

Each gallery piece you build will be its own fresh construction. Starting with a blank canvas ensures your practice shapes don't clutter your first real piece.

Can't find the Transform and Construct menu?

Make sure you've selected an object first. The Transform and Construct menu only appears in the toolbar when something is selected on the canvas.

Help me find the Transform and Construct menu.

Explore the expression list keyboard

The expression list is where you'll type precise transformation functions later in the project. Let's find where those functions live.

Click on an empty line in the expression list (left panel) to activate the Desmos keyboard at the bottom left.
Browse the functions listed under the Geometry Tools heading.
Browse the functions under the Measurements heading.
Browse the functions under the Transformations heading. This is where you'll find rotate, dilate, reflect, and translate functions.

Why does the expression list matter?

The toolbar lets you apply transformations by clicking. The expression list lets you define transformation functions you can reuse and combine.

In the Secret Mission, you'll type functions like R(x) = rotate(x, (0,0), 30) to build animated spirals. For now, just know where to find these tools.

Your workspace is ready. Next up, you'll build your first gallery piece: a parallel lines and transversals explorer that reveals angle relationships in real time.

Build the Parallel Lines Explorer

You've explored the Desmos Geometry interface and practiced the basic tools. Now it's time to build your first gallery piece.

This construction explores what happens when a straight line (called a transversal) crosses two parallel lines. You'll measure all 8 angles formed and watch their relationships hold no matter how you drag things around.

In this step, get ready to:

Construct two parallel lines using translation and add a transversal.
Measure and label all 8 angles at the two intersection points.
Verify angle relationships by dragging the transversal to different positions.

Construct two parallel lines and a transversal

A translation moves a shape in one direction without rotating or resizing it. Translating a line guarantees the copy is perfectly parallel to the original.

Select the Line tool from the geometry toolbar at the top.
Draw a horizontal line across the canvas by clicking two points that are roughly level with each other.
Switch to the selection tool (the arrow cursor) and click on the line to select it.
Open the Transform and Construct menu in the toolbar.
Choose Translate.
Click a start point on or near the line, then click a second point directly above it to set the translation direction.

A second line appears above the first. Because you created it with a vertical translation, these two lines are guaranteed to be parallel.

Select the Line tool again.
Draw a third line at an angle so it crosses both parallel lines. This is your transversal.

Why use Translate instead of drawing a second line freehand?

If you draw two separate lines by hand, they might look parallel but they are not mathematically linked. Desmos does not treat two freehand lines as parallel.

Translation creates a copy that is definitionally parallel to the original. This guarantees the angle relationships will hold perfectly when you measure them.

Don't see the Transform and Construct menu?

Make sure you selected the line first (click it with the arrow/selection tool). The menu only appears when an object is selected.

Help me find the Transform and Construct menu in Desmos Geometry.

Mark intersection points and measure all 8 angles

When a transversal crosses two parallel lines, it creates 4 angles at each intersection point for a total of 8 angles. Measuring all 8 reveals the hidden relationships between them.

Select the Intersect tool from the toolbar (or simply click where the transversal crosses a line).
Mark the intersection point where the transversal meets the first parallel line.
Mark the intersection point where the transversal meets the second parallel line.

You should see two distinct points, one on each parallel line where the transversal crosses it.

Select the Angle tool from the toolbar.
At the first intersection point, click three points to measure each of the 4 angles formed (click a point on one ray, then the vertex, then a point on the other ray).
Repeat at the second intersection point to measure all 4 angles there as well.

Measuring all 4 angles at one point

At each intersection, you have 4 angles to measure. Work around the point systematically (top-left, top-right, bottom-right, bottom-left) so you do not miss any.

Label each angle by selecting it and using the label option in the Selected Items Menu.
Give each angle a short label (like 1 through 8, or a through h) so you can identify them easily.

You should now see 8 angle measurements displayed on your canvas, 4 at each intersection.

Angle measurements showing 0 or not appearing?

Make sure you are clicking three distinct points when measuring: a point on one ray, then the vertex (intersection point), then a point on the other ray. Clicking the same point twice gives a zero-degree angle.

Help me measure angles correctly in Desmos Geometry.

Explore the angle relationships by dragging

This is where your construction comes alive. Dragging the transversal changes every angle measurement, but the relationships between angles stay locked in place.

Switch to the selection tool (arrow cursor).
Drag one of the points that define the transversal to a new position so the transversal tilts at a different angle.
Watch the 8 angle measurements update in real time.

What angle relationships should you see?

No matter where you drag the transversal, three relationships always hold:

Corresponding angles stay equal. These are angles in the same position at each intersection (e.g. both top-left angles).
Alternate interior angles stay equal. These are angles on opposite sides of the transversal, between the parallel lines.
Co-interior angles (same-side interior) always sum to 180 degrees.

Drag the transversal to at least 3 different positions and confirm each relationship still holds.
Save this construction by clicking the save button (you may need to sign in to your free Desmos account if you have not already).

Why do these relationships hold?

These angle relationships are a direct consequence of the two lines being parallel. Because you built the second line as a translation of the first, parallelism is guaranteed by construction. The angle equalities and supplementary pairs follow from the geometry of parallel lines cut by a transversal.

Angles not staying equal when you drag?

If corresponding angles are not staying equal, you may have drawn the second line freehand instead of using Translate. Undo your work back to the first line and redo the translation step. The second line must be created as a translation of the first.

Help me create parallel lines using translation in Desmos Geometry.

Your first gallery piece is complete. You have a dynamic proof that parallel lines create predictable angle relationships. Next up, you'll construct a triangle that reveals three hidden center points and a surprising line that connects them all.

Build the Triangle Centers Construction

Your parallel lines explorer is saved and ready to share. Now you will uncover something hidden inside every triangle.

Every triangle has special "center" points that appear when you construct specific lines (medians, perpendicular bisectors, and altitudes). In this step, you will plot three of these centers and discover a surprising line that connects all three of them.

In this step, get ready to:

Construct a scalene triangle and find its centroid by drawing medians.
Plot the circumcenter and orthocenter using perpendicular bisectors and altitudes.
Reveal the Euler Line connecting all three centers.

Draw the triangle and construct the centroid

A median is a line segment from a vertex to the midpoint of the opposite side. The three medians of any triangle always meet at a single point called the centroid.

Open a fresh Desmos Geometry canvas by navigating to desmos.com/geometry in a new browser tab.
Select the Polygon tool from the geometry toolbar.
Click three points on the graph paper to form a triangle, then click back on the starting point to close it.
Make the triangle scalene (all three sides different lengths) so the center points do not overlap each other.

Why scalene?

In equilateral or isosceles triangles, some center points sit on top of each other. A scalene triangle keeps all three centers visibly separated so you can see the Euler Line clearly.

Find the midpoint of the first side by clicking an empty expression line in the expression list and typing midpoint(). Position your cursor inside the parentheses, then hold Shift and click the first side of your triangle on the graph paper.
Repeat for the second and third sides so you have three midpoints.
Select the Segment tool and draw a segment from each vertex to the midpoint of the opposite side. These are the three medians.
Click the point where all three medians meet. This intersection is the centroid.
Select the centroid point, then click the label option in the Selected Items Menu to show its coordinates. Rename the label to G.

What is the centroid?

The centroid (G) is the triangle's balance point. If you cut the triangle out of cardboard, it would balance perfectly on a pin placed at G. It always divides each median in a 2:1 ratio from vertex to midpoint.

Construct the circumcenter and orthocenter

The circumcenter is where the three perpendicular bisectors meet. The orthocenter is where the three altitudes meet.

Select the first side of the triangle using the select tool.
Open the Transform and Construct menu in the toolbar.
Choose Perpendicular Bisector.
Repeat for the second and third sides. You now have three perpendicular bisectors.
Click the point where all three perpendicular bisectors intersect. This is the circumcenter.
Label this point O using the label option in the Selected Items Menu.

What is the circumcenter?

The circumcenter (O) is equidistant from all three vertices. You could draw a circle centered at O that passes through every vertex of the triangle. That circle is called the circumscribed circle.

Now construct the altitudes. Select the first vertex of the triangle.
Open the Transform and Construct menu and choose Perpendicular to the opposite side.
Repeat from each of the other two vertices to their opposite sides. You now have three altitudes.
Click the point where all three altitudes meet. This is the orthocenter.
Label this point H.

What is the orthocenter?

The orthocenter (H) is the intersection of the triangle's three altitudes. An altitude drops from a vertex perpendicular to the opposite side. Unlike the centroid, the orthocenter can land outside the triangle if the triangle is obtuse.

Reveal the Euler Line

A mathematician named Leonhard Euler discovered that G, O, and H always lie on the same straight line, no matter what triangle you draw. This is called the Euler Line.

Select the Line tool from the geometry toolbar.
Click on point G, then click on point O to draw a line through both.
Observe that point H also sits on this line. All three points are collinear.

Measure the distance from O to G by typing distance() in a new expression line, then Shift+clicking points O and G.
Measure the distance from G to H the same way in another expression line.
Drag any vertex of the triangle around the canvas. Watch how all three centers move but always stay on the same line.
Compare the two distance values. Notice that GH is always exactly twice OG, no matter where you drag the vertices.

Why is GH always twice OG?

This 2:1 ratio is a proven property of the Euler Line. The centroid always divides the segment from O to H in the ratio 1:2 (closer to O). Most geometry students do not encounter this result until university-level mathematics, so you are genuinely getting ahead.

Save this construction by clicking the share or save button in the top menu. This is your second gallery piece.

Points not lining up?

If H does not sit on the line through G and O, one of your altitudes may be slightly off. Delete the altitudes and re-draw them, making sure you select the correct vertex and choose Perpendicular to the opposite side each time.

If two of your centers overlap, your triangle is not scalene. Drag a vertex until all three sides are clearly different lengths.

Help me figure out why my three centers are not collinear.

You now have two gallery pieces saved. Next up, you will turn a single polygon into a full symmetrical art piece using reflections, rotations, and dilations.

Build the Transformation Art Piece

Your Euler Line construction proved that three seemingly unrelated triangle centers always line up. Now you are going to shift from discovery to design.

In this step, you will build a visually striking symmetrical design using nothing but rigid transformations and a single polygon. Every copy in the final piece traces back to one original shape, so dragging a single vertex reshapes the entire artwork.

In this step, get ready to:

Create an asymmetric polygon and color it as the base of your design.
Apply a reflection and multiple rotations to form a symmetrical ring.
Add a dilated inner ring and verify the entire design updates dynamically.

Create your base polygon

Every piece in your transformation art traces back to one original polygon. Making it asymmetric ensures that each reflected or rotated copy looks visibly different from the original, so the transformations are clearly visible.

Open a new blank canvas in Desmos Geometry for your third construction.
Select the Polygon tool from the geometry toolbar at the top.
Click 5 or 6 points on the graph paper near the center of the canvas to form an irregular shape. Make sure no two sides are the same length.
Click back on your starting point to close the polygon (or double-click the last vertex to auto-close it).
Click on your polygon to select it. In the Selected Items Menu that appears, open the style options and choose a color you like.

Why asymmetric?

If your polygon were a perfect square or equilateral triangle, some rotated copies would overlap and look identical. An irregular shape makes every transformation copy visually distinct, so you can clearly see the effect of each reflection and rotation.

Apply a reflection and rotations to form a ring

A reflection creates a mirror image across a line, and a rotation spins a shape around a fixed point by a given angle. By rotating in 60-degree increments, you get 6 evenly spaced copies that form a complete 360-degree ring.

Select the Line tool and draw a vertical line through the center of the canvas. This is your line of reflection.
Click on your polygon to select it.
Open the Transform and Construct menu in the toolbar.
Choose Reflect, then click the vertical line you just drew.

You should see a mirror image of your polygon appear on the other side of the vertical line.

Now select the original polygon again.
Open the Transform and Construct menu and choose Rotate.
Click the center point of the canvas as your center of rotation.
Type 60 into the angle input box and confirm.

You now have a rotated copy 60 degrees from the original. Next, repeat this rotation to fill the ring.

Select the rotated copy you just created.
Open the Transform and Construct menu. You should see your 60-degree rotation listed under Recent Transformations.
Click the recent rotation to apply it again. This creates a third copy at 120 degrees.
Repeat this process: select the newest copy, reuse the rotation from Recent Transformations. Continue until you have 6 copies forming a complete ring around the center point.

Why 60 degrees?

360 divided by 6 equals 60. Each copy lands exactly one-sixth of a full turn from the last, so 6 copies fill the circle perfectly with no gaps or overlaps.

Copies not forming a clean ring?

Make sure you are always choosing the same center point for each rotation. If the center shifts, copies will orbit around different spots.

Also confirm the angle is exactly 60 each time. A typo like 6 or 600 produces unexpected results.

Help me fix my rotation ring in Desmos Geometry

Add a dilated inner ring and test the dynamic updates

A dilation shrinks or enlarges a shape from a center point by a scale factor. A scale factor less than 1 shrinks. You will use this to create a smaller inner ring that adds depth to the design.

Select all 6 rotated copies. You can click each one while holding Shift, or drag a selection box around them.
Open the Transform and Construct menu and choose Dilate.
Click the center point of the canvas as your center of dilation.
Type 0.5 into the scale factor input box and confirm.

You should see a smaller set of 6 copies appear closer to the center, forming an inner ring.

Select the inner ring copies and change their color in the style options so they contrast with the outer ring.
Now drag one of the vertices of your original polygon. Watch the entire design (reflection, outer ring, and inner ring) update in real time.

Why does everything update together?

Every shape in your design is defined as a transformation of the original polygon. Desmos tracks these relationships live. When you move the original, every dependent copy recalculates its position automatically.

This is the power of geometric constructions over static drawings. The relationships are built in, not just drawn.

Inner ring not updating when you drag?

Make sure you are dragging a vertex of the original polygon (the first one you drew), not a vertex of one of the copies. Only the original drives the chain of transformations.

If the inner ring did not appear at all, confirm you selected all 6 outer copies before applying the dilation. Missing even one means it will not be included in the inner ring.

Help me fix my dilation in Desmos Geometry

Save this construction to your Desmos account. This is your third gallery piece.

Your transformation art piece proves that reflections, rotations, and dilations can turn one simple shape into complex symmetrical art. Next up, you will build your final gallery piece: an interactive explorer that proves a powerful theorem about inscribed angles in circles.

Build the Inscribed Angle Theorem Explorer

Your transformation art piece shows how a single polygon can generate an entire symmetrical design. Now you'll build the final gallery piece. This one proves a theorem most geometry students don't encounter until later in the course.

The Inscribed Angle Theorem says that any angle inscribed in a circle is always exactly half the central angle that shares the same arc. In this step, you'll construct an interactive explorer that makes this relationship impossible to miss.

In this step, get ready to:

Draw a circle and construct a central angle with a measured value.
Create an inscribed angle on the same arc and measure it.
Verify the 2:1 ratio dynamically by dragging points around the circle.

Draw the circle and mark the central angle

A central angle is formed by two segments that both start at the center of a circle and end on the circumference. You'll draw this first so you have something to compare the inscribed angle against.

Start a new blank canvas in Desmos Geometry (open desmos.com/geometry in a new tab if your previous construction is still open).
Select the Circle tool from the geometry toolbar at the top.
Click once on the canvas to place the center point.
Click a second time away from the center to set a point on the circumference and complete the circle.

Why two clicks?

The first click defines the center. The second click defines the radius by placing a point on the circumference. The circle tool draws the full circle through that second point.

Select the Point tool from the toolbar.
Click directly on the circle's circumference to place a point. This is point A.
Click on a different spot on the circumference to place a second point. This is point B.
Select the Segment tool from the toolbar.
Draw a segment from the center point to point A.
Draw a second segment from the center point to point B.
Select the Angle tool from the toolbar.
Click point A, then the center, then point B to measure the central angle.

You should see an angle measurement displayed at the center of the circle between the two segments.

Don't see an angle measurement?

Make sure you clicked exactly three points in order: A, center, B. The middle click must be the vertex of the angle (the center point). If the measurement appears outside the circle, undo and try clicking the points in the opposite order (B, center, A).

Help me measure the central angle in Desmos Geometry.

Place a point on the arc and measure the inscribed angle

An inscribed angle has its vertex on the circle itself (not at the center). Its two sides are segments that connect the vertex to the same two points (A and B) that define the central angle.

Select the Point tool.
Click on the circumference to place point C. Make sure C is on the major arc (the larger arc that does not sit directly between A and B).
Select the Segment tool.
Draw a segment from point C to point A.
Draw a segment from point C to point B.
Select the Angle tool.
Click point A, then point C, then point B to measure the inscribed angle ACB.

You should see a second angle measurement at point C. Compare it to the central angle measurement at the center.

What should I see?

The inscribed angle at C should be exactly half of the central angle at the center. For example, if your central angle reads 80 degrees, the inscribed angle should read 40 degrees.

Inscribed angle not showing half the central angle?

Double check that point C is on the major arc (the arc that does NOT contain the shorter path between A and B). If C is on the minor arc, the relationship flips and the angles won't show the expected 2:1 ratio.

Also confirm that your angle measurement at C uses the correct vertex. The middle click must be point C (the vertex of the inscribed angle).

Help me fix my inscribed angle measurement.

Verify the theorem dynamically and save

The real power of this construction is that it proves the theorem is not a coincidence for one specific arrangement. It holds for every possible position of C on the arc.

Click the pointer tool (the arrow at the top left of the toolbar) to switch to selection mode.
Click and drag point C along the major arc of the circle.
Watch both angle measurements as you drag. The inscribed angle stays exactly half the central angle no matter where C sits on the major arc.
Now drag point A to a new position on the circumference.
Observe that the central angle changes, and the inscribed angle updates to remain exactly half.
Try dragging point B as well to confirm the 2:1 ratio holds for any configuration.

Why does this always work?

The Inscribed Angle Theorem is a fundamental property of circles. Any inscribed angle that subtends ("opens toward") the same arc as a central angle will always measure exactly half. This is true regardless of where the vertex sits on the remaining arc.

This is one of those results that feels surprising until you see it proven visually. Your construction does exactly that.

Label your points by selecting each one and using the label option in the Selected Items Menu. Label them A, B, and C.
Save this construction by pressing Cmd+S (macOS) or Ctrl+S (Windows/Chromebook).

Your fourth and final gallery piece is complete. You now have a full set of interactive geometry constructions.

Having trouble saving?

If you are not signed in, Desmos will prompt you to create a free account or sign in with Google. Sign in to save your constructions so you can access them later from any browser.

Help me save my Desmos Geometry construction.

Secret mission

Animated Spiral Art with Composite Transformations

What if you could turn a single shape into an infinite spiral that animates in real time? In this secret mission, you'll combine rotation and dilation into a composite transformation function, then drive it with a slider to create mesmerizing animated spiral art.

Clean Up Your Resources

This project runs entirely in your browser using Desmos Geometry, which is free. There are no ongoing costs and nothing expires.

Resources you used:

Parallel Lines and Transversals Explorer (saved Desmos Geometry construction).
Triangle Centers and Euler Line construction (saved Desmos Geometry construction).
Transformation Art construction (saved Desmos Geometry construction).
Inscribed Angle Theorem Explorer (saved Desmos Geometry construction).
Animated Spiral (Secret Mission, saved Desmos Geometry construction).
Desmos account (free).

Keep everything running

No action needed. Choose this if you want your gallery ready to show classmates or your teacher at any time.

Your constructions live in your Desmos account indefinitely.
Open them anytime by visiting desmos.com/geometry and clicking the folder icon in the top left.
Share any construction by clicking the share button in the top right corner to copy its unique link.

Pause - I'll come back to this later

No action needed. Since everything is saved to your free Desmos account, your constructions persist whether the browser is open or closed.

Close your browser tabs. Your constructions remain in your account.
To return later, visit desmos.com/geometry and click the folder icon (or press Cmd+O on macOS or Ctrl+O on Windows/Chromebook) to see your saved constructions.

Delete - I don't want to use this again

Remove all your gallery constructions from your Desmos account. This cannot be undone after 5 seconds.

In Desmos Geometry, click the folder icon in the top left (or press Cmd+O on macOS or Ctrl+O on Chromebook) to open your Saved Work.
Check the box next to each of your five gallery constructions.
Click Delete Items to remove them all at once.
Your Desmos account itself is free and does not need to be deleted.

Nice Work!

You just built a full Geometric Art Gallery in Desmos Geometry. Four interactive constructions that prove real theorems visually, all shareable from your Chromebook.

You've learned how to:

Built interactive angle relationship explorers proving corresponding, alternate interior, and co-interior angle relationships with parallel lines, and proving the Inscribed Angle Theorem dynamically.
Built a Triangle Centers construction revealing the centroid, circumcenter, orthocenter, and the Euler Line (GH = 2 times OG).
Built a Transformation Art piece using reflections, rotations, and dilations of a single polygon into a symmetrical design.
Secret Mission: Built an animated spiral using composite transformation functions and a slider, combining algebraic function definition with geometric construction.

Ready to quiz yourself?