$$PQ$$ is a diameter of circle and $$XY$$ is chord equal to the radius of the circle. $$PX$$ and $$QY$$ when extended intersect at $$E$$. Prove that $$\angle PEQ = 60^\circ$$.

Note by Vishwathiga Jayasankar
2 years, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. 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 1paragraph 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 $$2 \times 3$$
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} $$\boxed{123}$$

Sort by:

Let $$O$$ denote circle center. It can be shown in general that $$\angle PEQ = \angle OXY=\angle OYX$$ regardless of $$XY$$ length and if it is parallel to $$PQ$$ or not.

$$\angle OPX=\angle OXP, \angle OQY=\angle OYQ$$

$$\angle E = 180-\angle OPX - \angle OQY = 180-\angle OXP-\angle OYQ=180 - \angle YXE - \angle XYE$$

$$(180-\angle OXP - \angle YXE) +(180-\angle OYQ-\angle XYE)=2 \angle OXY = 2 \angle E \Rightarrow$$

$$\angle E=\angle OXY$$

- 2 years, 3 months ago

- 2 years, 3 months ago

Are PQ and XY parallel?

- 2 years, 3 months ago

I at first thought that might be a necessary condition, but after looking at several orientations for $$XY$$ it appears that $$\angle PAQ = 60^{\circ}$$ in general, which would be an interesting result.

- 2 years, 3 months ago

So it will be 60° even if PQ and XY are not parallel ?

- 2 years, 3 months ago

Yes, I haven't determined a proof yet, but that result does seem to hold in general.

- 2 years, 3 months ago