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Side Ratio Resolution (Posted on 2015-07-18) Difficulty: 3 of 5
Triangle PQR has incentre I and circumcentre O. Suppose that ∠ PIO = 90o and ∠ QIO = 45o.

Find the ratio PQ: QR : RP

No Solution Yet Submitted by K Sengupta    
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re(2): Solution | Comment 3 of 4 |
(In reply to re: Solution by Jer)


I don't know if this construction
will help you with a proof for the
problem.

Geometer's Sketchpad Construction:

1. Construct a fixed segment IP.

2. Construct line m through I 
   perpendicular to IP.

3. Construct point O on line m.

4. Construct ray IE such that
   angle PIQ = 135 degrees and
   point O lies in its interior.

5. Construct circle C with center O
   and passing through point P.
   
6. Point Q is the intersection of
   circle C and ray IE.

7. Reflect line PQ about IP to 
   get line PS.

8. Reflect line PQ about IQ to
   get line PT.

9. Point R is the intersection of
   PS and PT ( Note angle PRQ is
   90 degrees ).

10. Note as point O moves on line m
    that point R is inside circle C
    when O is inside triangle PQR
    and point R is outside circle C
    when O is outside triangle PQR.

    R lies on circle C when O is
    the midpoint of segment PQ.

  Posted by Bractals on 2015-07-20 17:27:26
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