How to show two lengths are equal

There are many ways to show two segments are of equal length. Let us first discuss such problems. There are too many theorems and results which can be used, but let us just list a few more common techniques:

  • Using congruent triangles;

  • Using a third line segment;

  • Using isosceles triangles;

  • Using parallelograms;

  • Circle Properties: Chords of equal length are of the same distance from the center of the circle, chords subtended by the same angle at the center of the circle are of the same length, etc;

  • If a line parallel to one side of a triangle bisects a second side, it also bisects the third side;

  • Comparing the lengths with multiples or 12\frac{1}{2}, 13\frac{1}{3} ... of another length.

Exercises

11. Construct squares ABEFABEF and ACGHACGH externally to the sides ABAB and ACAC respective of triangle ABCABC. Let DD be the foot of the perpendicular from AA to BCBC, and let ADAD intersect FHFH at MM. Show that FM=MHFM=MH.

22. In ABC\triangle{ABC}, AB=ACAB=AC. Let DD be a point on ABAB and EE be a point on ACAC extended such that BD=CEBD=CE. Let DEDE intersect BCBC at FF. Show that DF=FEDF=FE.

33. Let EE and FF be the midpoints on the sides BCBC and ADAD respectively of a parallelogram ABCDABCD. Show that BFBF and DEDE trisect ACAC.

44. Let ABCABC be a triangle. Construct equilateral triangles ABDABD and ACEACE externally to sides ABAB and ACAC respectively. Let FF be the point such that ADFEADFE is a parallelogram. Show that FBC\triangle{FBC} is equilateral.

55. Let BDBD and CECE be the altitudes of a triangle ABCABC. Let FF be the midpoint of BCBC and GG be on DEDE such that DEFGDE \perp FG. Show that DG=GEDG=GE.

Note by Victor Loh
5 years, 2 months ago

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