Angles of the triangle

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Angles of the triangle

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

Source: May Olympiad(Olimpiada de Mayo) 2005

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

647 posts

#1 h Feb 21, 2018, 11:43 PM • 2 Y

Y by Adventure10, Mango247

In a triangle $ABC$ with $AB = AC$ , let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$ . The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$ . Find the angles of the triangle $BAC$ .

This post has been edited 2 times. Last edited by mathisreal, Feb 21, 2018, 11:43 PM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

372 posts

#2 h Feb 22, 2018, 3:50 AM • 2 Y

Y by Adventure10, Mango247

mathisreal wrote:

In a triangle $ABC$ with $AB = AC$ , let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$ . The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$ . Find the angles of the triangle $BAC$ .

Assume that $\angle BAD=k$ , then $\angle DAM=2k$ and $\angle MAC=3k$ , also $\angle ACB=90-3k$ .
Consider $\triangle ACD$ , assume that $\overline{AM}$ intersects $\overline{NC}$ at $E$ .
We can see that $E$ is the orthocenter of $\triangle ACD$ . But $\overline{DN}=\overline{DM}$ , so $\triangle ACD$ is an isosceles triangle with $\overline{DA}=\overline{DC}$ .
We conclude that $5k=90-3k$ , $k=\frac{45}{4}$ . So $\angle BAC=6k=67.5$
Done

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a