P, Q and R are three points located on a circle L with diameter 4 and satisfying PQ = QR. Point S is located inside L in such a manner that QR = RS = SQ. The line passing through P and S intersects L at the point T.
Determine the length of ST.
Using trigonometry.
Let the center be point C.
Let angle PCQ = QCR = 2a degrees, where a <= 30 degrees (this is the first case)
1) Triangle PCQ is isosceles, so, angle PCQ = PQC = 90 - a
2) Triangle QRS is equilateral, so angle RQS = SRQ = 60 degrees
3) Angle CRS = CQS = CQR - SQR = 30 - a
4) Angle PQS = PQC + CQS = 120 - 2a
5) Triangle PQS is isoceles, so angle SPQ = PSQ = 30+a
6) Angle CPT = CPQ - SPQ = 60 - 2a
7) Triangle CTP is isoceles, so CTP = CPT = 60 - 2a,
and PCT = 60 + 4a
8) Angle CTR = PCT - PCQ - QCR = 60 degrees!
(Now we're getting someplace)
9) CTR is isosceles, and CTR = 60 degrees,
so triangle CTR is equilateral and
RT = the radius of the circle = 2!!
10) Angle SRT = SRC + CRT = 90 - a
11) Angle RST = 180 - PTR - SRT = 90 - a
12) So STR is isoceles, and ST = RT = 2 !!!
There might be something simpler, but I am tired and won't find it.
And I'm sure I can make an exactly parallel argument if a > 30 degrees, but I'm not going to.
Isoceles power
