Use the points on the circles and/or their centres to form isosceles and equilateral triangles. The circles are of the same size.

Question

Homi · Accepted Answer

Understand the Question

All given circles are of equal radius $r$ .
When the center of one circle lies on the circumference of another, the distance between their centers is equal to the radius $r$ .
Any line segment connecting a circle's center to a point on its circumference has length $r$ .
Isosceles triangle: A triangle with at least two equal sides.
Equilateral triangle: A triangle with all three sides equal (note that every equilateral triangle is also isosceles).

Step 1 · Establish Radii and Distances Between Centers

Let the radius of each identical circle be $r$ . Diagram 1

In the two-circle diagram, circle $A$ has center $A$ and passes through $B$ , so $AB = r$ . Circle $B$ has center $B$ and passes through $A$ , so $BA = r$ .
In the three-circle diagram, each circle passes through the centers of the other two, giving: $AB = BC = CA = r$

Step 2 · Form Isosceles Triangles

An isosceles triangle requires at least two equal sides.

In the two-circle configuration, where circles intersect at points $X$ and $Y$ :

In $\Delta AXY$ : $AX = AY = r$ Since two sides are equal, $\Delta AXY$ is an isosceles triangle.
In $\Delta BXY$ : $BX = BY = r$ Since two sides are equal, $\Delta BXY$ is an isosceles triangle.

Step 3 · Form Equilateral Triangles

An equilateral triangle requires all three sides to be equal.

From the two-circle configuration:

From the three-circle configuration:

Connecting the three centers in $\Delta ABC$ : $AB = BC = CA = r \implies \Delta ABC \text{ is equilateral}$
For the other intersection points $P$ (intersection of circles $A$ and $B$ ), $Q$ (circles $B$ and $C$ ), and $R$ (circles $C$ and $A$ ): $PA = PB = AB = r \implies \Delta PAB \text{ is equilateral}$ $QB = QC = BC = r \implies \Delta QBC \text{ is equilateral}$ $RA = RC = AC = r \implies \Delta RAC \text{ is equilateral}$

(Note: Every equilateral triangle also satisfies the condition for being an isosceles triangle.)

Answer

Isosceles Triangles: $\Delta AXY$ , $\Delta BXY$ (and all equilateral triangles)

Equilateral Triangles: $\Delta AXB$ , $\Delta AYB$ , $\Delta ABC$ , $\Delta PAB$ , $\Delta QBC$ , and $\Delta RAC$

Common Mistakes

Overlooking the Distance Between Centers: Missing that because circle $A$ passes through center $B$ , the distance $AB$ is also equal to the radius $r$ .
Separating Isosceles and Equilateral Categories: Forgetting that all equilateral triangles are technically isosceles as well, since they have at least two equal sides.

Question 2