2022 SSMO Team Round Problems/Problem 5

Problem

Consider the following rectangle $ABCD$ where $BC=8.$ If $CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,$ find the value of $[P_1CP_2A].$ (Note that $[ABC]$ is the area of $ABC$ .)

$[asy] unitsize(0.25cm); dot((0,0)); dot((12,0)); dot((12,16)); dot((0,16)); draw((0,0)--(12,0)--(12,16)--(0,16)--(0,0)--cycle); dot((8,22)); dot((17,4)); dot((8,16)); dot((12,4)); label("$A$", (0,0), SW); label("$B$", (12,0), SE); label("$C$", (12,16), NE); label("$D$", (0,16), NW); label("$P_1$", (8,22), NE); label("$P_2$", (17,4), SE); label("$T_1$", (8,16), SE); label("$T_2$", (12,4), SW); draw((8,22)--(8,16),dashed+linewidth(.5)); draw((12,4)--(17,4),dashed+linewidth(.5)); [/asy]$

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2022 SSMO Team Round Problems/Problem 5

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