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ABC and PQR altitude angle ABC assignable base called circumference circumscribed polygons corresponding diameter difference distances divided draw drawn drawn parallel equal equal angles equimultiples equivalent feet figure finding the greatest fixed point four magnitudes fourth geometric given circle given point given straight lines greater greatest common measure half harmonic identical in form known length less Let ABC let the angle line joining magnitudes meet multiple number of sides once parallel parallelogram pencil perimeter placed portions process of finding produced proof proportional Prove ratio rectangle BK rectangle contained rectilineal figure regular remainder respectively right angle segments Shew sides similar polygons square Suppose taken THEOREM third three straight lines transversal triangle ABC triangle PQR twice
Page 150 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Page 178 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion...
Page 174 - To find a mean proportional between two given straight lines. Let AB, BC be the two given straight lines; it is required to find a mean proportional between them. Place AB, BC in a straight line, and upon AC describe the semicircle ADC, and from the point B draw* BD at right angles to AC: n * 11.
Page 155 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.
Page 146 - The locus of a point whose distances from two fixed points are in a constant ratio (not one of equality) is a circle.
Page 149 - THEOREM. 43. 7\ťo triangles are equal, when the three sides of the one are equal to the three sides of the other, each to each.
Page 147 - Each of the three straight lines which join the angular points of a triangle to the middle points of the opposite sides is called a Median of the triangle.
Page 143 - If an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.
Page 151 - Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles...
Page 148 - THEOR. If the sides of two triangles, about each of their angles, be proportionals, the triangles shall be equiangular, and have their equal angles opposite to the homologous sides. Let the triangles ABC, DEF have their sides proportionals, so that AB is to BC, as DE to EF ; and BC to CA, as EF to FD ; and consequently, ex...