A baffling geometry question

In the given figure, P and Q are point of contact. O is the incentre. Line BO produced, meets PQ at G. Find the value of angle AGB?

The reason i shared the question as a note is that i don't know the answer. Try it out, & if you solve, please post a rigorous solution rather than mere guess.

Note by Sanjeet Raria
5 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Check out this graphic

Baffling Geometry Baffling Geometry

With lots of angle-chasing, it can be worked out that, given angles a,ba, b, the following angles are

OAP=90ab\angle OAP=90-a-b
OGP=90+a+b\angle OGP=90+a+b
AOG=90a\angle AOG=90-a
APG=90+a\angle APG=90+a

Hence, APGOAPGO is a cyclic quadrilateral. Moreover, since APOAPO is a right triangle, AOAO is the diameter of the circle that circumscribes the cyclic quadrilateral APGOAPGO. Hence, AGOAGO is also a right triangle, and we have our answer.

Michael Mendrin - 5 years, 1 month ago

Log in to reply

Here is another way to prove it. It is easy to see PQC=BAC/2+ABC/2 \angle PQC = {\angle BAC} /2 + {\angle ABC}/2 . So you can get BGQ=BAC/2=BAO \angle BGQ = {\angle BAC} /2 = \angle BAO, therefore BGQBAO \triangle BGQ \sim \triangle BAO and BG×BO=BQ×BA BG \times BO = BQ \times BA . Let the point of contact of incircle with AB is D, we have BG×BO=BD×BA BG \times BO = BD \times BA . So AGOD is cyclic. AGB=BDO=90 \angle AGB = \angle BDO = 90 ^ \circ

Roger Lu - 5 years, 1 month ago

Log in to reply

You haven't defined the point DD in this proof. But it is simply the point touching the circle on the segment ABAB.

mathh mathh - 5 years, 1 month ago

Log in to reply

AP PG GB

Shanzkie Vargas - 5 years, 1 month ago

Log in to reply

Construct a model diagram with AC and BC AB as tangents

führer sy - 5 years ago

Log in to reply

With any dimensions of radius and lengths of triangle

führer sy - 5 years ago

Log in to reply

I took an equilateral triangle and using coordinate geometry found out the value of the angle. Is the method legitimate?

Kartik Raj - 5 years, 1 month ago

Log in to reply

You've only found it when the triangle is equilateral. You haven't proved that the angle is the same for all triangles. You can't assume that all the triangles have the same angle just because the question asks for a single unique numeric value.

mathh mathh - 5 years, 1 month ago

Log in to reply

90

führer sy - 5 years ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...