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ABC and PQR altitude angle ABC base BC as PQ bisected CD as FG circumference circumscribed polygons corresponding diameter differences of multiples drawn parallel equal angles equi equimultiples finding the greatest fixed points four magnitudes given circle given point given ratio given straight lines greater than qmD greatest common measure harmonic mean homologous identical in form inscribed and circumscribed Let ABC let the angle line joining lines be proportional magnitudes be proportional MANCHESTER GRAMMAR SCHOOL mean proportional number of sides parallel to AP parallelogram pentagon perimeter PQ to QR process of finding prove quadrilateral figure ratio are equal rectangle BK rectangle contained rectilineal figure regular polygons remainder right angle right-angled triangle segments similar polygons similar triangles similarly described square Theorem third three straight lines tiples triangle ABC triangle PQR
Pagina 152 - 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'.
Pagina 180 - 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...
Pagina 176 - 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.
Pagina 157 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.
Pagina 148 - The locus of a point whose distances from two fixed points are in a constant ratio (not one of equality) is a circle.
Pagina 151 - THEOREM. 43. 7\Ľo triangles are equal, when the three sides of the one are equal to the three sides of the other, each to each.
Pagina 149 - 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.
Pagina 145 - 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.
Pagina 153 - 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...
Pagina 150 - 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...