(con)cyclic wanted, projections of vertices on medians related

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Source: 2014 Saudi Arabia GMO TST day I p1

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#1 h Jul 31, 2020, 12:51 PM • 1 Y

Y by dchenmathcounts

Let $ABC$ be a triangle with $\angle A < \angle B \le \angle C$ , $M$ and $N$ the midpoints of sides $CA$ and $AB$ , respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$ , respectively. Prove that the quadrilateral $MNPQ$ is cyclic.

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#2 h Jul 31, 2020, 9:31 PM • 2 Y

Y by Mathlete2017, Mango247

The given angle conditions imply no configuration issues. Let medians $BM$ and $CN$ intersect at centroid $R$ . Note that we wish to prove $RP\cdot RN = RQ \cdot RM$ . Since $\angle CQB = \angle CPB=90^{\circ}$ , $CBPQ$ is cyclic. PoP with respect to the circumcircle of $CBPQ$ , yields $RP = \frac{RQ\cdot RB}{RC}$ and $RQ = \frac{RP\cdot RC}{RB}$ . Thus we wish to prove $\frac{RQ\cdot RB}{RC}\cdot RN = \frac{RP\cdot RC}{RB} \cdot RM \implies RB^2\cdot RQ \cdot RN = RC^2 \cdot RP \cdot RM.$ Note that $\triangle CRQ \sim \triangle BRP\implies \frac{CR}{RQ} = \frac{BR}{RP}$ , thus we wish to prove $RB\cdot RN = RC \cdot RM$ . This is true, since $RN = \frac{RC}{2}$ and $RM=\frac{RB}{2}$ . $\blacksquare$

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#3 h Jul 31, 2020, 11:10 PM • 1 Y

Y by Mathlete2017

Thanks for the good problems, parmenides51 - I swear half of my geometry handouts would be barren of examples/exercises if it weren't for the problems you post.

Solution

I just want to note that the weird angle conditions do not matter - my solution (and GeoGebra) do not care about configurations.

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#4 h Aug 1, 2020, 8:54 PM • 3 Y

Y by Mathlete2017, dchenmathcounts, Inconsistent

all the problems that I post, are always mentioned their source (either a magazine or a contest)
there are a few problems that I have posted without source where the aops' admins are to blame for losing the source as they moved them from HSO to HSM and the source field was lost (as HSM has no source field)

I am glad that someone uses them, to provide a different problem set from the well known problems, as many Russian Constests are a lot worthy

Edit: If there is any problem posted in aops by me sourceless, just send me the link (PM) to search in my page for the correct source

This post has been edited 3 times. Last edited by parmenides51, Aug 2, 2020, 2:46 PM

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