UKMT Special (Problem $12$ )

The diagram shows a triangle $ABC$ with points $T, U$ on the edge $AB$ , points $P , Q$ on $BC$ and $R, S$ on $CA$ , where:

$SP$ and $AB$ are parallel, $UR$ and $BC$ are parallel and $QT$ and $CA$ are parallel

$SP, UR$ and $QT$ all pass through a point $Y$ .

$PQ = RS = TU$

Prove that

$\frac{1}{PQ} = \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$

[UKMT Maclaurin Olympiad $2018$ , M $6$ ]

#Geometry

Easy Math Editor

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Comments

You have until $2^{nd}$ December, $3:00$ pm!

Yajat Shamji - 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

UKMT Special (Problem 121212)

Comments

UKMT Special (Problem $12$ )