fun circle geometry question (1 Viewer)

[EvoRevolution]

A cyclic quadrilateral has perpendicular diagonals which meet in P.
Prove that the feet of the perpendiculars from p to the sides of the quadrilateral lie on a circle.

 

[MC Squidge]

ill draw a pic so we can git the ball rollin

 

[MC Squidge]

ohhhhh gee wiz guys i only drew the first part of the question feel free to add some

 

[gurmies]

Hmm, are these perpendicular feet at 90 degrees to the sides of the original quadrilateral?

 

[EvoRevolution]

ohhhhh gee wiz guys i only drew the first part of the question feel free to add some

lool

 

[MC Squidge]

if some1 can interperet the question and finnish my pic i can solve it since i am prety gud with the circle rules

 

[Timothy.Siu]

this prove that the red thing is a cyclic quad

 

[gurmies]

if some1 can interperet the question and finnish my pic i can solve it since i am prety gud with the circle rules

Not to mention highly modest

 

[MC Squidge]

ohhh that ones a thinker give me a minute

 

[gurmies]

It's a kite at the very least. Could be a rhombus or square...

 

[GUSSSSSSSSSSSSS]

It's a kite at the very least. Could be a rhombus or square...

yep but had NO idea wat i wud do form there..

 

[Iruka]

Classic case of converse fallacy.

A kite has diagonals that cross at right angles, but that does not mean that any quad with perpendicular diagonals is a kite...

 

[Drongoski]

I've solved it. But without being able (unlike jetblack2007) to put up the diagram it's not practical to post soln.

Soln outline:

1) The 4 perpendiculars fr P to the 4 sides (see Timothy.Siu's diagram) partition the original cyclic quad into 4 cyclic quads

2) Then I worked out ( haha) 2 opp angles of the quad formed by the 4 feet of perpendiculars sum to 180 deg.

3) Had used repeatedly: Angles on a common chord in a segment are equal to these smaller cyclic quads. Also 2 of them add to 90 deg.

 

[Iruka]

Explain in brief how you did it.

I had lots of little cyclic quads inside the big one.

 

[EvoRevolution]

I've solved it. But without being able (unlike jetblack2007) to put up the diagram it's not practical to post soln.

draw in paint lol.

 

[Drongoski]

draw in paint lol.

u don't understand. I'm quite illiterate in use of computers. I barely use a mobile phone. had to get someone to help me retrieve an sms message. That's how pathetic it is!

That's why my ability to post solutions is handicapped.

 

[Iruka]

u don't understand. I'm quite illiterate in use of computers. I barely use a mobile phone. had to get someone to help me retrieve an sms message. That's how pathetic it is!

That's why my ability to post solutions is handicapped.

The solution is not too hard to do on a bit of scrap paper if you just angle chase a bit, but it would be a pain to write it all out.

 

[Drongoski]

Indeed!

 

[EvoRevolution]

u don't understand. I'm quite illiterate in use of computers. I barely use a mobile phone. had to get someone to help me retrieve an sms message. That's how pathetic it is!

That's why my ability to post solutions is handicapped.

lol


anyway ill give it a try