Calendar: 1855-1856 Page 414

410 THE SCHOOL SECOND CLASS MATHEMATICAL SCHOLARSHIP I-Qiulftf To describe an equilateral triangle upon given finite straight line If the sides CA CB be produced to meet the circles in and the line FG will pass through the second point of inter- section of the circles and the triangle CFG times CAB In any right-angled triangle the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle Provethat£ i2 A'A'2 AC BC2 To describe an isosceles triangle having each of the angles at the base double of the third angle If DC Β be produced to meet the circle again in and and EF be joined then Δ CEF Δ ABB χ Γ2 In every triangle the square of the side subtending either of the acute angles is less than the squares of the sides containing that angle by twice the rectangle contained by either of these sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle From the extremity of the base BC of the Δ ABC draw Β Ε bisecting Cm and from draw AD perpendicular to Β or BC produced then shall- BE2 CE BC BD If from point without circle there be drawn two straight lines one of which cuts the circle and the other meets it if the rectangle contained by the whole line which cuts the circle and the part of it without the circle be equal to the square of the line which meets it the line which meets shall touch the circle 10 ABC is an equilateral triangle Ο the centre of the inscribed circle through draw AEOD to the base cutting the circle in then AD shall be trisected in Ο and 11 If two triangles have one angle of the one equal to one angle of the other and the sides about the equal angles propor- tionals the triangles shall be equiangular and shall have those angles equal which are opposite to the homologous sides 12 If through any point in the triangle ABC lines Aa Βb Co be drawn to the opposite sides then shall- Ac Β Cb aC bA cB