Triangle PQR has incentre I and circumcentre O. Suppose that ∠ PIO = 90^{o} and ∠ QIO = 45^{o}.
Find the ratio PQ: QR : RP
(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 20150720 17:27:26 