Paraguayan National Olympiad 2010, Level 3, Problem 5

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0 replies

jlacosta

Apr 2, 2025

0 replies

MONT pg 31 example 1.10.40

Jaxman8 0

30 minutes ago

Can somebody explain why it works.

0 replies

Jaxman8

30 minutes ago

0 replies

NEPAL TST 2025 DAY 2

Tony_stark0094 9

N 31 minutes ago by hectorleo123

Consider an acute triangle $\Delta ABC$ . Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$ , respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$ . Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$ , and by $Q$ the intersection of ray $EB$ with $\Gamma$ .

If $O$ is the circumcenter of $\Delta ABC$ , prove that $O$ , $D$ , and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$

9 replies

Tony_stark0094

Apr 12, 2025

hectorleo123

31 minutes ago

lines CV, BU intersect on the circumcircle of ABC

parmenides51 4

N 37 minutes ago by ihategeo_1969

Source: 2019 Geo Mock - Olympiad by Tovi Wen #4 https://artofproblemsolving.com/community/c594864h1787237p11805928

Let $ABC$ be a triangle whose incircle is tangent to $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ at $D, E, F$ , respectively. Let $M$ and $N$ be the midpoints of $\overline{DE}$ and $\overline{DF}$ , respectively. Suppose that points $U$ and $V$ lie on $\overline{MN}$ so that $BU = NU$ and $CV = MV$ . Prove that lines $\overline{CV}$ and $\overline{BU}$ intersect on the circumcircle of $\triangle ABC$ .

4 replies

parmenides51

Nov 26, 2023

ihategeo_1969

37 minutes ago

one cyclic formed by two cyclic

CrazyInMath 33

N 39 minutes ago by breloje17fr

Source: EGMO 2025/3

Let $ABC$ be an acute triangle. Points $B, D, E$ , and $C$ lie on a line in this order and satisfy $BD = DE = EC$ . Let $M$ and $N$ be the midpoints of $AD$ and $AE$ , respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$ , respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.

33 replies

CrazyInMath

Apr 13, 2025

breloje17fr

39 minutes ago

No more topics!

Paraguayan National Olympiad 2010, Level 3, Problem 5

Leicich 2

N Aug 3, 2021 by ike.chen

In a triangle $ABC$ , let $D$ , $E$ and $F$ be the feet of the altitudes from $A$ , $B$ and $C$ respectively. Let $D'$ , $E'$ and $F'$ be the second intersection of lines $AD$ , $BE$ and $CF$ with the circumcircle of $ABC$ . Show that the triangles $DEF$ and $D'E'F'$ are similar.

2 replies

Leicich

Sep 23, 2014

ike.chen

Aug 3, 2021

Paraguayan National Olympiad 2010, Level 3, Problem 5

G H J

Reveal topic tags

geometry

circumcircle

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Leicich

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Leicich

#1 h Sep 23, 2014, 10:34 PM • 3 Y

Y by jhu08, Adventure10, Mango247

This post has been edited 1 time. Last edited by Leicich, Apr 26, 2018, 3:57 AM
Reason: Removed comment from top-level post

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Talisman

735 posts

Talisman

#2 h Sep 24, 2014, 1:58 PM • 3 Y

Y by jhu08, Adventure10, Mango247

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$\angle CBE=\angle CAD=\angle CBD',\triangle BDH\equiv\triangle BDD',DH=DD',$ similarly, $EH=EE',FH=FF'$
$DE\parallel D'E',EF\parallel E'F',FD\parallel F'D',\triangle DEF\sim\triangle D'E'F'.$

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ike.chen

1162 posts

ike.chen

#3 h Aug 3, 2021, 6:04 PM • 1 Y

Y by jhu08

What Is This Problem

Let $H$ be the orthocenter of $ABC$ . By the Orthocenter Reflection Lemma, we know $HD' = 2 \cdot HD, HE' = 2 \cdot HE, HF' = 2 \cdot HF.$ Now, the result follows by taking a homothety at $H$ with $k = 2$ . $\blacksquare$

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