SECTION II
Marks
6. P is a point on the side GH of the triangle FGH. Points Q
and R are taken on FH and FG respectively such that GQ
and FIR intersect on FP. QR cuts GH at S. By applying
the theorems of Ceva and Menelaus, prove that P and S
divide GH internally and externally in the same ratio. (5)
In the triangle ABC, the excircle opposite to A touches BC,
CA, AB at P, Q, R respectively. Prove that AP, BQ, and CR
are concurrent. (5)
If QR cuts BC at L, RP cuts CA at M, and PQ cuts AB at N,
prove that L, M, and N are collinear and that the line of
collinearity is the polar with respect to the excircle of the
point at which AP, BQ, and CR are concurrent. (10)
7. Prove that the arms of an angle and its internal and external
bisectors form a harmonic pencil. (6)
AB is a chord of a circle and CD the diameter at right-angles
to AB. P is any point on the circumference of the circle.
(i) Prove that PC and PD harmonically separate PA and PB. (4)
(ii) If, further, CD cuts AB in M, and X and Y are the feet of
the perpendiculars from A and B respectively to CP, prove
that MY is parallel to AP and that MX and MY
harmonically separate MB and MP. (5) 5)
8. Define inverse points and prove that any circle drawn
through a pair of points which are inverse with respect to a
given circle is orthogonal to the given circle.
(i) A and B are the centres of two unequal and non-intersect¬
ing circles. HK is a direct common tangent. The circle
whose diameter is HK cuts AB in X and Y. Prove that X
and Y are inverse points with respect to each of the circles
centres A and B.
Another circle, S, passes through X and Y and cuts the
circle with centre A at P and Q and the circle with centre B
at R and S. PQ and RS cut at L. Prove that LX and Lh
are tangents to the circle S.
(ii) Given a fixed circle, S, and two fixed points, P and Q,
not on a diameter of S, state how to construct a circle
orthogonal to S and passing through P and Q.
(5)
(4)
(6)
(5)
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