Some Noteworthy Properties of the Triangle And Its Circles
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A = = -- = -- 22 2 > b + c 44.
45.
46. * = 4A 47. R = - -\/bcs(s-a).
48.
r " = __ j a J b s c (For proofs of 43-50, see any text. ) 49. In an acute triangle, / +/' +/" = R + r.
Proof.
C Let ABC (Fig. 11) be a A, Z>, E, F mid-points of its sides, P the circumcentre. Denote BC by a, AC by b, AB by c, PE by /, PF by /', by p'.
26 The Triangle BDPE is a cyclic quadrilateral. Then BE x PD + BD x PE = DE x (The product of the diagonals of an inscribed quadrilateral is equal to the sum of the product ...of its opposite sides. ) Or, or, ap'" + cp' = Z>R. (i) Similarly, ap" + bp' = cR, (2) and bp'" + cp"=aR. (3) Again, ap' + bp" + cp" = 2 A = ar + br + cr. (4) Adding and factoring, (a + + ^)(/ +/' +/") = ( + b + ;)(tf + r).
Reducing, p' +p" +/" =R + r.
50. In case of an obtuse triangle, p 1 " being drawn to longest side, / +p" p"' = R + r.
51 I + i+-L = l.
W A * . | It* III yi yii Y n
What to read after Some Noteworthy Properties of the Triangle And Its Circles?
You can find similar books in the "Read Also" column, or choose other free books by W H William Herschel Bruce to read online
To quickly assess the difficulty of the text, read a short excerpt:
A = = -- = -- 22 2 > b + c 44.
45.
46. * = 4A 47. R = - -\/bcs(s-a).
48.
r " = __ j a J b s c (For proofs of 43-50, see any text. ) 49. In an acute triangle, / +/' +/" = R + r.
Proof.
C Let ABC (Fig. 11) be a A, Z>, E, F mid-points of its sides, P the circumcentre. Denote BC by a, AC by b, AB by c, PE by /, PF by /', by p'.
26 The Triangle BDPE is a cyclic quadrilateral. Then BE x PD + BD x PE = DE x (The product of the diagonals of an inscribed quadrilateral is equal to the sum of the product ...of its opposite sides. ) Or, or, ap'" + cp' = Z>R. (i) Similarly, ap" + bp' = cR, (2) and bp'" + cp"=aR. (3) Again, ap' + bp" + cp" = 2 A = ar + br + cr. (4) Adding and factoring, (a + + ^)(/ +/' +/") = ( + b + ;)(tf + r).
Reducing, p' +p" +/" =R + r.
50. In case of an obtuse triangle, p 1 " being drawn to longest side, / +p" p"' = R + r.
51 I + i+-L = l.
W A * . | It* III yi yii Y n
What to read after Some Noteworthy Properties of the Triangle And Its Circles?
You can find similar books in the "Read Also" column, or choose other free books by W H William Herschel Bruce to read online
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