...trisection of each of them. 217. To construct a triangle having given the three medians. 218. The three perpendiculars from the vertices of a triangle on the opposite sides are concurrent. 219. The lines perpendicular to the sides of a triangle through their middle points are concurrent....

...position between AD and AF ; AE lies in magnitude between AD and AF. I. 19, Cor. 20. Since the sum of the perpendiculars from the vertices of a triangle on the opposite sides is greater than the semiperimeter, by the fifteenth deduction ; and since, by the preceding deduction,...

...(Professor Malet, FR8.)— If A = I—**2, prove that * loc1 A -^7332. (The Editor.) — If plt p,, p, be the perpendiculars from the vertices of a triangle on the opposite sides ; rf,, dj, rfa the distances from the vertices to the points of contact of the escribed circles with...

...from B or C on the opposite side of the triangle meet AD at O ; shew that OD is equal to DP. *i66. The perpendiculars from the vertices of a triangle on the opposite sides meet in a point (called the orthocentre of the triangle). *i67. If O is the orthocentre of the triangle...

...the mean point of eu e2, e3, and trisects the distance from el to p^ (2) To find the common point of the perpendiculars from the vertices of a triangle on the opposite sides. Let l, m, n be the ratios of the sides of the triangle to the cosines of the opposite angles ; ie T>f*...

...= -. (3) If л/o2 — x¡i cos ß + o sin a = x sin /3, find a. (4) If Z, m, и, be the lengths of the perpendiculars from the vertices of a triangle on the opposite sides, prove that the area of the triangle is - Z' »»' и* (cosec A cosec В cosec C)*. ¿ * (5) An object...

...PQR divides the line joining the median points of the triangles ABC, A'B'C' in the ratio m : n. 115. The perpendiculars from the vertices of a triangle on the opposite sides meet in a point (§ 9G, Ex. 2), which is called the orthocentre of the triangle. If ABC be the triangle,...

...Find the condition that the vectors x — a, b—c may be perpendicular, and hence show that the three perpendiculars from the vertices of a triangle on the opposite sides are concurrent. 18119. (VV SATYANARAYAN.) — The ratio compounded of the ratios in which any two i»ogonal conjugates...

...10. Show that the perpendicular bisectors of the sides of a triangle meet in a point. 11. Show that the perpendiculars from the vertices of a triangle on the opposite sides meet in a point. 12. Show that the three points of intersection found in Problems 9, 10, and 11 lie...

...the curve. Prove that the centre of the circle is on the ellipse 21. The co-ordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are (20, 25), (8, 16) and (8, 9). Find the co-ordinates of the vertices of the triangle. Am. Any three...