Question

Triangle $ABC$ has a right angle at $B$ and contains a point $P$ for which $PA=10,$ $PB=6,$ and $\angle APB=\angle BPC=\angle CPA.$ Find $PC.$

Answer

Answer: 33

Concepts:vector right triangle

Difficulty rating: 2380

Small Hint:

The three equal angles around $P$ are each $120^\circ$

Big Hint:

Use vectors from $P$ and translate the right angle at $B$ into a dot product

Solution:

Let $\mathbf a,$ $\mathbf b,$ and $\mathbf c$ be the vectors from $P$ to $A,$ $B,$ $C,$ and put $|\mathbf c|=x.$ Their pairwise angles are $120^\circ,$ so $\mathbf a\cdot\mathbf b=-30,$ $\mathbf b\cdot\mathbf c=-3x,$ and $\mathbf a\cdot\mathbf c=-5x.$ Since $AB\perp BC,$

$(\mathbf a-\mathbf b)\cdot(\mathbf c-\mathbf b)=0.$ Expanding gives $-5x+30+3x+36=0,$ so $x=33.$

1987 AIME Problem 9

Answer: 33

Difficulty rating: 2380

Small Hint:

Big Hint:

Solution:

Problem 9 in Other Years