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OlympiadBench / 1953 / Let ABC be an acute triangle, and let M be the midpoint of AC. A circle…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Numerical

is multiple answer

false

language

English

question

Let

A B C

be an acute triangle, and let

M

be the midpoint of

A C

. A circle

\omega

passing through

B

and

M

meets the sides

A B

and

B C

again at

P

and

Q

, respectively. Let

T

be the point such that the quadrilateral

B P T Q

is a parallelogram. Suppose that

T

lies on the circumcircle of the triangle

A B C

. Determine all possible values of

B T / B M

Plain-text mathematical notation (without MathML)

Let ABC be an acute triangle, and let M be the midpoint of AC. A circle ω passing through B and M meets the sides AB and BC again at P and Q, respectively. Let T be the point such that the quadrilateral BPTQ is a parallelogram. Suppose that T lies on the circumcircle of the triangle ABC. Determine all possible values of BT/BM.

Original LaTeX notation

Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triangle $A B C$. Determine all possible values of $B T / B M$.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1953
task_id: 853e0017-f251-5f0f-bbe2-acf94d66eb00
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1953
task_revision_id: 2
config / split: OE_TO_maths_en_COMP / train

Official source

initial import