you then drop a perpendicular from B to D.
this creates two other triangles
they are:
right triangle ADB where angle ADB is the right angle.
right triangle BDC where angle BDC is the right angle.
in triangle ABC, if you assume that angle BCA is equal to x, then angle BAC must be equal to 90 - x because the acute angles in a right triangle are complementary.
in right triangle ADB, since angle DAB is the same angle as angle BAC in right triangle ABC, then angle DAB is equal to 90 - x and angle DBA is equal to x.
in right triangle BDC, since angle DCB is the same angle as angle ACB in triangle ABC, then angle DCB is equal to x and angle DBC is equal to 90 - x.
what this all means is that:
triangle ABC is similar to triangle ADB which is similar to triangle BDC
while difficult to see in diagram 1, this is eaier to see in diagrams 2 and 3 and 4 where the similar triangle have been oriented so that their corresponding sides and corresponding angles are line up the same way.
since these triangle are similar, then their corresponding sides are proportional.
that similarity relationship is what creates the geometric mean theorem.
what the geometric mean theorem states is based on diagram 1 in the picture.
first rule is that AD / AB = AB / AC
this rule is based on the similarity relationships in diagrams 2 and 3.
AB in diagram 2 and AD in diagram 3 are corresponding sides of similar triangles.
AC in diagram 2 and AB in diagram 3 are corresponding sides of similar triangles.
since the triangle are similar, the corresponding sides are in proportion and you get:
AD in diagram 3 is to AB in diagram 2 as AB in diagram 3 is to AC in diagram 2.
algebraically, this is shown as:
AD / AB = AB / AC
second rule is that DC / BC = BC / AC
this rule is based on the similarity relationships in diagrams 2 and 4.
BC in diagram 2 and DC in diagram 4 are corresponding sides of similar triangles.
AC in diagram 2 and BC in diagram 4 are corresponding sides of similar triangles.
since the triangle are similar, the corresponding sides are in proportion and you get:
DC in diagram 4 is to BC in diagram 2 as BC in diagram 4 is to AC in diagram 2.
algebraically, this is shown as:
DC / BC = BC / AC
the third rule is that AD / BD = BD / CD
this rule is based on the similarity relationships in diagrams 3 and 4.
DB in diagram 3 and DC in diagram 4 are corresponding sides of similar triangles.
DA in diagram 3 and DB in diagram 4 are corresponding sides of similar triangles.
since the triangle are similar, the corresponding sides are in proportion and you get:
DA in diagram 3 is to DB in diagram 4 as DB in diagram 3 is to
DC in diagram 4
algebraically, this is shown as:
DA / DB = DB / DC
since DA is the same as AD and DB is the same as BD and DC is the same as CD, the ration can also be shown as:
AD / BD = BD / DC
the diagrams are shown below:
the rules and how they are visualized in diagram 1 are shown below:
