Math Gold Medalist

Lor

2023 AIME 

Problem 8

Rhombus $ABCD$ has $\angle BAD < 90^\circ.$ There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to the lines $DA,AB,$ and $BC$ are $9,5,$ and $16,$ respectively. Find the perimeter of $ABCD.$

Congruent Triangles

Pythagorean Theorem


Draw 4 perpendiculars from P to AB, BC, CD, DA and call them K, L, F, E respectively.

Prove that

PF=20

r=25/2

Suppose the incircle touches AB, BC, CD, DA  at N, Q, J, M respectively.

Prove that in the following steps:

EM=12

KN=10

AK=15

AE=13

LT=12

FJ=10

CQ=25

CJ=25

LQ=12

BQ=25/4

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   Solution