Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/409

 and having its acute angle at A: if AF be joined, shew that the angle BAC is, divided into three equal angles.

480. AB, AC are two fixed straight lines at right angles; D is any point in AB, and E is any point in AC; on DE as diagonal a half square is described with its vertex at G: shew that the locus of G is the straight line which bisects the angle BAC.

481. Shew that a square is greater than any other parallelogram of the same perimeter.

482. Inscribe a square of given magnitude in a given square.

483. ABC is a triangle; AD is a third of AB, and AE is a third of AC; CD and BE intersect at F: shew that the triangle BFC is half the triangle BAC, and that the quadrilateral ADFE is equal to either of the triangles CFE or BDF.

484. ABC is a triangle, having the angle C a right angle; the angle A is bisected by a straight line which meets BC at D, and the angle B is bisected by a straight line which meets AC' at E; AD and BE intersect at 'O: shew that the triangle AOB is half the quadrilateral ABDE.

485. Shew that a scalene triangle cannot be divided by a straight line into two parts which will coincide.

486. ABCD, ACED are parallelograms on equal bases BC, CE, and between the same parallels AD, BE; the straight lines BD and AE intersect at F: shew that BF is equal to twice DF.

487. Parallelograms AFGC, CBKH are described on AC,BC outside the triangle ABC; FG and KH meet at Z; ZC is joined, and through A and B straight lines AD and BE are drawn, both parallel to ZC, and meeting FG and KH at D and E respectively: shew that the figure ADEB is a parallelogram, and that it is equal to the sum of the parallelograms FC, CK.

488. If a quadrilateral have two of its sides parallel shew that the straight line drawn parallel to these sides through the intersection of the diagonals is bisected at that point.

489. Two triangles are on equal bases and between the same parallels: shew that the sides of the triangles intercept equal lengths of any. straight line which is parallel to their bases.