Interesting Configuration with Semicircle

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0 replies

jlacosta

Apr 2, 2025

0 replies

Fixed angle

iv999xyz 0

29 minutes ago

Let ABC be an acute triangle with circumcircle w and D is fixed point on BC. E is randomly selected on BC and C, D and E are in this order. AE intersects w again at F and circumcircle around DEF intersects w again at M. Prove that ∠ABM is fixed.

0 replies

+1 w

iv999xyz

29 minutes ago

0 replies

6 tangents to 1 circle

moony_ 1

N 41 minutes ago by soryn

Source: own

Let $P$ be a point inside the triangle $ABC$ . $AP$ intersects $BC$ at $A_0$ . Points $B_0$ and $C_0$ are defined similarly. Line $B_0C_0$ intersects $(ABC)$ at points $A_1$ , $A_2$ . The tangents at these points to $(ABC)$ intersect BC at points $A_3$ , $A_4$ . Points $B_3$ , $B_4$ , $C_3$ , $C_4$ are defined similarly. Prove that points $A_3$ , $A_4$ , $B_3$ , $B_4$ , $C_3$ , $C_4$ lie on one conic

1 reply

moony_

Yesterday at 9:30 AM

soryn

41 minutes ago

2024 numbers in a circle

PEKKA 30

N 42 minutes ago by NicoN9

Source: Canada MO 2024/2

Jane writes down $2024$ natural numbers around the perimeter of a circle. She wants the $2024$ products of adjacent pairs of numbers to be exactly the set $\{ 1!, 2!, \ldots, 2024! \}.$ Can she accomplish this?

30 replies

PEKKA

Mar 8, 2024

NicoN9

42 minutes ago

Inspired by ZDSX 2025 Q845

sqing 5

N an hour ago by sqing

Source: Own

Let $a,b,c>0$ and $a^2+b^2+c^2 +ab+bc+ca=6$ . Prove that $\frac{1}{2a+bc }+ \frac{1}{2b+ca }+ \frac{1}{2c+ab }\geq 1$

5 replies

sqing

Yesterday at 1:41 PM

sqing

an hour ago

No more topics!

Interesting Configuration with Semicircle

hakN 3

N Nov 4, 2024 by tranducphat

Source: Turkey EGMO TST 2022 P5

We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$ , and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$ . If the line $RS$ meets the extension of the diameter at $Z$ , prove that $ZC$ is tangent to the semicircle.