Geometry

I want a problem about circles, tangents, secants and inscribed angles. Please create some questions and problems! thank you!

#Geometry #Circles #TangentOfCircles

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Here's a problem made to order for you, it has "circles, tangents, secants and inscribed angles" See graphic.

I Want A Problem

Two circles $1$ and $2$ are tangent at point $A$ . Points $(P, R)$ are on circle $1$ . A secant is drawn through points $(P,R)$ , and tangents are drawn through point $P$ and $R$ . These tangents intersect circle $2$ at points $Q$ and $S$ . Another secant is drawn through points $(Q,S)$ , which intersects the other secant at point $B$ . Two more secants are drawn through points $(Q,A)$ and points $(S,A)$ . Prove that the line drawn through points $(A,B)$ bisects the angle between those two secants. That is, prove that the inscribed angles the two secants through $(Q,A)$ and $(S,A)$ make with the line through $(A,B)$ are equal.

Michael Mendrin - 6 years, 8 months ago

Michael, I hope you haven't given him more than he can handle ;)

A Former Brilliant Member - 6 years, 8 months ago

Maybe I should come up with another problem with half as many things, you know, a problem about a circle, a tangent, a secant, and an inscribed angle. It's not easy coming up with a problem with multiples of each.

How about if you posted a solution for me to another one of my geometry problems? This one

You can guess but can you prove it?

Everybody's "solving" this one, but nobody has actually put up a full proof.

Michael Mendrin - 6 years, 8 months ago

@Michael Mendrin – Hehe I didn't read the note at the bottom of the question. Ooops. Back to the whiteboard.

A Former Brilliant Member - 6 years, 8 months ago

Okay, here's a problem made with fewer circles, tangents, secants, etc. See graphic

Geo Problem

Points $A,B,C$ lie on a circle. A tangent is drawn through point $A$ . Secants are drawn through points $(A,B)$ and points $(B,C)$ . The two secants intersect at point $D$ . Prove that these two inscribed angles are equal $\angle DAC=\angle ABC$

Michael Mendrin - 6 years, 8 months ago

Check out the problems in Circles - Problem Solving - Basic and Circles - Problem Solving - Intermediate.

Calvin Lin Staff - 6 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$