Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/421

 figure, we can form a similar rectilineal figure circumscribing the circle.

596. Find a mean proportional between two similar right-angled triangles which have one of the sides containing the right angle common.

597. in the sides AC, BC of a triangle ABC points D and E are taken, such that CD and CE are respectively the third parts of AC and BC; BD and AE are drawn intersecting at O: shew that EO and DO are respectively the fourth parts of AE and BD.

598. CA, CB are diameters of two circles which touch each other externally at C; a chord AD of the former circle, when produced, touches the latter at E, while a chord BF of the latter, when produced, touches the former at G: shew that the rectangle contained by AD and BE is four times that contained by DE and FG.

599. Two circles intersect at A, and BAG is drawn meeting them at B and C; with B, C as centres are described two circles each of which intersects one of the former at right angles: shew that these circles and the circle whose diameter is BC meet at a point.

600. ABCDEF is a regular hexagon: shew that BF divides AD in the ratio of one to three.

601. ABC, DEF are triangles, having the angle A equal to the angle D; and AB is equal to DF: shew that the areas of the triangles are as AC to DE.

602. If M, N be the points at which the inscribed and an escribed circle touch the side AC of a triangle ABC; shew that if BM be produced to cut the escribed circle again at P, then NP is a diameter.

603. The angle A of a triangle ABC is a right angle, and D is the foot of the perpendicular from A on BC; DM, DN are perpendiculars on AB, AC: shew that the angles BMC, BNC are equal.

604. If from the point of bisection of any given arc of a circle two straight lines be drawn, cutting the chord of the arc and the circumference, the four points of intersection shall also lie in the circumference of a circle.

605. The side AB of a triangle ABC is touched by the inscribed circle at D, and by the escribed circle at E: shew that the rectangle contained by the radii is equal to the rectangle AD, DB and to the rectangle AE, EB.