Question on circles

Let $ ABC$ be a triangle with circumcentre $O$. The points $P$ and $ Q $are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $ M $be the mid points of the segments $BP,CQ$ and $PQ$ respectively, and let $\tau$ be the circle passing through $K,L $and $ M$.Suppose that the line $PQ$ is tangent to the circle $\tau$.Prove that $OP=OQ$.

#Geometry #Proofs #MathProblem #Math

Easy Math Editor

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Comments

QP tangents with the small circle at M, we have ∠QMK = ∠MLK.

M, K and L are midpoints of PQ, BP and QC, respectively; therefore,

KM || QB, KM = ½ QB(a), ML || PC, ML = ½ PC(b)

and ∠QMK = ∠MQA, or ∠MLK = ∠MQA.

ML || PC and KM || QB therefore ∠QAP = ∠KML

The two triangles QAP and KML are similar since their respective angles are equal. Therefore, ML/QA = KM/AP

From (a) and (b) AP × PC = QA × QB(c)

Extend PQ and QP to meet the larger circle at U and V, respectively.

In the larger circle UV intercepts AB at Q, we have

QU × QV = QA × QB or QU × (QP + PV) = QA × QB (i)

UV intercepts AC at P, we have UP × PV = AP × PC or (QU + QP) × PV = AP x PC (ii)

From (i) and from (c) QU × (QP + PV) = AP × PC

Therefore, from (ii) QU × (QP + PV) = (QU + QP) × PV

Or PV = QU and M is also the midpoint of UV and OM ⊥UV

Therefore OP = OQ

Anubhav Singh - 7 years, 7 months ago

this is my favourite question

Anubhav Singh - 7 years, 5 months ago

Hmm ... ..ho gaya.

Ujjwal Mani Tripathi - 7 years, 8 months ago

HOW???

Ankush Tiwari - 7 years, 8 months ago

Dono bhai 95% pakar bhag gaye....mil jao kisi din bahut marenge.... :)

Anubhav Singh - 6 years, 11 months ago

add a new status :)

Anubhav Singh - 7 years, 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}`	$\boxed{123}$