equal segments in an isosceles inscribed trapezoid

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Source: Nordic Mathematical Contest 2002 #1

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#1 h Sep 24, 2017, 9:29 AM • 2 Y

Y by Adventure10, Mango247

The trapezium ${ABCD}$ , where ${AB}$ and ${CD}$ are parallel and ${AD < CD}$ , is inscribed in the circle ${c}$ . Let ${DP}$ be a chord of the circle, parallel to ${AC}$ . Assume that the tangent to ${c}$ at ${D}$ meets the line ${AB}$ at ${E}$ and that ${PB}$ and ${DC}$ meet at ${Q}$ . Show that ${EQ = AC}$ .

This post has been edited 1 time. Last edited by parmenides51, Sep 24, 2017, 11:45 AM
Reason: latex added

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GGPiku

#2 h Sep 24, 2017, 9:53 AM • 2 Y

Y by Adventure10, Mango247

Let $BP$ intersect the tangent at $X$ ., We have $AD=CB=PC$ , so $DC$ is the angle bisector of $\angle BDP$ . $\angle XDP=\angle PBD$ , $\angle PDQ=\angle QDB$ , so $XDQ$ is isoscelles, $DQ\parallel EB$ , $DQBE$ is an isoscelles trapezium, $\angle QCA=\angle DBA=\angle QEB$ , so $QCAE$ is a parallelogram, hence $EQ=AC$ .

This post has been edited 1 time. Last edited by GGPiku, Sep 24, 2017, 9:53 AM

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