KL+MN=LM+KN, projections in cyclic ABCD, (2011 Kukin MO by OmSU 11.5)

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KL+MN=LM+KN, projections in cyclic ABCD, (2011 Kukin MO by OmSU 11.5)

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#1 h Apr 11, 2021, 5:47 PM • 2 Y

Y by samrocksnature, Mango247

The diagonals of the cyclic quadrilateral $ABCD$ intersect at point $X$ . Points $K, L, M, N$ are feet of perpendiculars drawn from point $X$ on sides $AB, BC, CD$ and $DA$ , respectively. Prove that $KL + MN = LM + KN$ .

(folklore)

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#2 h Apr 11, 2021, 5:57 PM • 1 Y

Y by samrocksnature

Is this just Ptolemy's + Pythag?

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#3 h Apr 11, 2021, 6:01 PM

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What revered culture is this that a geometry problem is part of their folklore?

@below ohh ic thx

This post has been edited 2 times. Last edited by samrocksnature, Apr 11, 2021, 6:18 PM

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#4 h Apr 11, 2021, 6:13 PM • 1 Y

Y by samrocksnature

in Russian Math olympiads, when a problem is known from the past (e.g. 40 years ago from an Olympiad), they use the term ''folklore''

Imagine that in Russia, they have too many City Olympiads, many math tournaments / battles, and many Rounds for many years, so they have a huge database of such problems

with math problems, 152 olympiads take place every year, imagine that many of them have more than 1 round, so ...

This post has been edited 1 time. Last edited by parmenides51, Apr 11, 2021, 6:14 PM

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Quantum_fluctuations

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Quantum_fluctuations

#5 h Apr 12, 2021, 6:00 AM • 2 Y

Y by samrocksnature, Mango247

parmenides51 wrote:

in Russian Math olympiads, when a problem is known from the past (e.g. 40 years ago from an Olympiad), they use the term ''folklore''

Imagine that in Russia, they have too many City Olympiads, many math tournaments / battles, and many Rounds for many years, so they have a huge database of such problems

with math problems, 152 olympiads take place every year, imagine that many of them have more than 1 round, so ...

WOW!!

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