Geometry Questions [[urgent]]?

Question

1. Show that if 3 divides n2 evenly then 3 divides n evenly. [Hint: n can only be of the form 3a, 3a + 1 or 3a + 2]
2. The point P lies within the angle ABC and is equidistant from AB and BC. Prove that PB bisects the 3.In triangle ABC, P is a point on BC such that PB PC and PA bisects

Answer 1

1.Since 3 is not a square, any square with 3 for a factor must, by definition, have 3*3 for a factor.Therefore if 3|(n^2), 3|n.

2. Let PD be the perpendicular distance from P to line AB and PE be the perpendicular distance from P to BC. It is given that PD = PE and AP equals itself, so ∆ABP is congruent to ∆CBP by Side-Side-Angle. Therefore angle ABP = angle PBC.

3. If BAP = PAC and PB = PC, with PA common to both triangles, ∆BAP is congruent to ∆PAC, again, by SSA. Therefore AB must equal AC.

Answer 2

Let the quotient of n^2/3 equal an integer. By inspection, this can be written as n (n/3). If n is an integer, the only way the product can be an integer is if n is divisible by 3.

2. I don't think this is provable.
3. You can prove that the angle bisector of an isoceles triangle, where the angle bisected is not one of the equal angles, is the bisector of opposite side. This is the converse. This is also the reason you can't prove 2 unless you stipulate that AB=BC.