*Problems Part 2

1. Two triangles are congruent if two sides and the enclosed median in one triangle are respectively equal to two sides and the enclosed median of the other.
2. The two sides AB, AC in the triangle ABC are produced to D, E respectively so that BD = BC = CE. If BE and CD intersect in F, show that ∠BFD = right angle –  ½∠A.
3. From any point D on the base BC of an isosceles triangle ABC, a perpendicular is drawn to cut BA and CA or produced in M and N. Prove that AMN is an isosceles triangle.
4. ABC is a triangle in which AB is greater than AC. If D is the middle point of BC, then the angle CAD will be greater than the angle BAD.
5. The straight line drawn from the middle point of the base of a triangle at right angles to the base will meet the greater of the two sides, not the less.
6. ABC is a right-angled triangle at A. The altitude AD is drawn to the hypotenuse BC. DA and CB are produced to P and Q respectively so that AP = AB and BQ = AC. Show that CP = AQ.
7. A square is described on the hypotenuse BC of a right-angled triangle ABC on the opposite side to the triangle. If M is the intersection of the diagonals of the square and LMN is drawn perpendicular to MA to meet AB, AC produced in L, N respectively, then BL = AC,CN = AB.
8. Show that the sum of the altitudes of a triangle is less than the sum of its three sides.
9. On BC as a base, an equilateral triangle ABC and an isosceles triangle BBC are drawn on the same side of BC such that ∠D = half ∠A.Prove that AB = BC.
10. P is any point inside or outside the triangle ABC. AP, BP, CP are produced to R, S, T respectively so that AP = PR, BP = PS, CP = PT. Show that the triangles RST and ABC are equiangular.
11. The interior and exterior angles at C of a triangle ABC are bisected by CD,CF to meet AB and BA produced in D, F respectively. From D a line BR is drawn parallel to BC to meet AC in R. Show that FR produced bisects BC. (Produce DR to meet CF in S.)
12. ABC is an isosceles triangle in which AB = AC. On AB a point G is taken and on AC produced the distance CH is taken so that BG = CH. Prove that GH > BC.
13. Construct a triangle having given the base, the difference of the base angles, and the difference of the other two sides. [P1-10]
14. Show that the sum of the three medians in a triangle is less than its perimeter and greater than ¾  the perimeter.
15. ABC is an isosceles triangle and D any point in the base BC; show that perpendiculars to BC through the middle points of BD and DC meet AB, AC in points H, K respectively so that BH = AK and AH = CK. (Join BH, BK.)
16. The side AB of an equilateral triangle ABC is produced to D so that BD = 2 AB. A perpendicular DF is drawn from D to CB produced. Show that FAC is a right angle.
17. On the two arms of a right angle with vertex at A, AB is taken = AD and also AC = AE, so that B, C are on the same area of ∠A. Prove that the perpendicular from A to CD when produced bisects BE.