Proof in circles and tangents?

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Homework Help Overview

The discussion revolves around a geometric proof involving a circle and tangents. The original poster presents a scenario with a circle (S1) and a point (P) outside of it, from which two tangents are drawn to touch the circle at points Q and R. The task is to prove that a circle passing through points P, Q, and R also passes through the center of circle S1.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of congruent triangles, specifically referencing right-angle, hypotenuse, and side relationships. There are inquiries about which specific triangles are being considered in the proof. Some suggest drawing lines from point P to the center of the circle and from the center to the points of tangency to identify relevant triangles. Others mention properties related to diameters and angles in circles.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants have offered insights into the geometric relationships at play, while others express confusion about the problem's requirements. There is no explicit consensus yet, but several lines of reasoning are being examined.

Contextual Notes

Participants are working within the constraints of a homework problem, and there is a sense of uncertainty about the proof's direction. The original poster acknowledges difficulty in understanding the problem and expresses gratitude for the responses received.

vkash
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there is a circle say S1,there is a point P outside circle. Two tangents are drawn from the point P to the circle. These two tangents touches circles S1 at Q and R. A circle is made through P,Q,R. proof that circle is passing through center of S1.

I have tried to put some triangles congruent similar and some other ideas but all failed.
 
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For starters, I see congruent triangles (Right-angle, Hypotenuse, Side).
 
NascentOxygen said:
For starters, I see congruent triangles (Right-angle, Hypotenuse, Side).

how?? which triangles??
 
You have tangents to a circle? Then there are your right-angles.
 
Draw the line from the given point to the center of the circle. Draw the radii from the center of the circle to the points of tangency. There are your triangles. Also use the fact that if two vertices of a triangle lie on the ends of a diameter of a circle and the third vertex is on the circle, then the angle at that third point is a right angle.
 
Obviously, the midpoint of S1P (let's call it T) is equidistant from both S1 & P (by definition of midpoint).

That is, TS1 = TP

All you need to do is first show that PQ = PR, then show that TP = TQ = TS1
 
this is so simple question. i did not got the point. thanks to all answerers.
sorry for late reply.
 

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