Equilateral Triangles

Try this

Drag $P$ : its distances to the three sides (orange, blue, green) each change. But the three-color bar on the right, which stacks the three pieces end to end, always reaches exactly as high as $A$ .
Drag $P$ onto a side: one piece becomes $0$ , and the other two still add up to the altitude. Drag $P$ to the vertex $A$ : only one piece is left, and it is the altitude.
Drag $P$ outside the triangle: the bar is now taller than the altitude. The result only holds for points inside (and on the sides).
The three colored small triangles $△ P B C$ , $△ P C A$ , $△ P A B$ fit together into the whole triangle. That is the reason.

Predict first

A point $P$ inside an equilateral triangle is at distances $1$ , $2$ , $3$ from the three sides. What is the altitude of the triangle?

Choose an answer first; then you can see the result.

Why: count the area twice

The distances to the three sides add up to the altitude

Step 1 / 4

Join P to the three vertices
The equilateral triangle is split into three pieces: $△ P B C$ , $△ P C A$ , $△ P A B$ .

$S_{△ A B C} = S_{△ P B C} + S_{△ P C A} + S_{△ P A B}$
Every piece has base a
The area of the whole
Cancel ½a

Rotating by 60°: joining three segments into one path

The other gift of the equilateral triangle: rotate it $6 0^{\circ}$ about any vertex and one side lands exactly on another. More generally, a $6 0^{\circ}$ rotation about a point creates an equilateral triangle out of nothing: the center of rotation, a point, and its image.

Let's use it on a famous problem: find the point $P$ in $△ A B C$ that makes $P A + P B + P C$ as small as possible.

First drag

P

yourself and push the sum as low as you can. Then click “Rotate △BPA 60° about B” and see where the three segments went.

When is PA + PB + PC smallest?

Step 1 / 5

Three segments spreading out from one point
$P A$ , $P B$ , $P C$ are like three spokes. They are not joined end to end, so “the shortest path between two points is a straight segment” does not apply.

$P A + P B + P C$
Rotate △BPA 60° about B
An equilateral triangle appears
The three pieces join into one path
Pull it straight

(If the triangle has an angle of $12 0^{\circ}$ or more, no such point exists, and the minimum is at that obtuse vertex. Try dragging $A$ down in the figure above.)

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: For every point inside, the sum of the distances to the three sides is constant

An equilateral triangle. Wherever $P$ is inside, the sum equals the altitude.

Practice

Exercise 1Basic

An equilateral triangle has altitude $9$ . A point $P$ inside it is at distances $2$ and $3$ from two of the sides. Find its distance to the third side.

Exercise 2Basic

An equilateral triangle has altitude $63$ . Find its side length.

Exercise 3Advanced

$P$ is a point inside the equilateral triangle $△ A B C$ with $P A = 3$ , $P B = 4$ , $P C = 5$ . Find $∠ A P B$ in degrees.

Exercise 4Challenge

On the outside of $△ A B C$ , build equilateral triangles $A B D$ on side $A B$ and $A C E$ on side $A C$ . Join $C D$ and $B E$ . What is the acute angle between them, in degrees?

Try this

Why: count the area twice

The distances to the three sides add up to the altitude

Rotating by 60°: joining three segments into one path

When is PA + PB + PC smallest?

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Using “the sum of distances is constant” in any triangle

Rotating without saying where the equilateral triangle comes from