MathDB

Four lighthouses are located at points

A

B

C

, and

D

. The lighthouse at

A

5

kilometers from the lighthouse at

B

, the lighthouse at

B

12

kilometers from the lighthouse at

C

, and the lighthouse at

A

13

kilometers from the lighthouse at

C

. To an observer at

A

, the angle determined by the lights at

B

and

D

and the angle determined by the lights at

C

and

D

are equal. To an observer at

C

, the angle determined by the lights at

A

and

B

and the angle determined by the lights at

D

and

B

are equal. The number of kilometers from

A

D

is given by

\displaystyle\frac{p\sqrt{r}}{q}

, where

p

q

, and

r

are relatively prime positive integers, and

r

is not divisible by the square of any prime. Find p\plus{}q\plus{}r,

Problem Statement