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AXYZ has medians AZ and BX. Draw YO so that it intersects segment XZ at C.
Construct segments HZ and HX such that H is on YO and HZ || XBand HX || BZ.
Complete the following steps of a paragraph proof to prove that YC meets the other two medians at O.
By the reflexive property of congruence, ZOYB ZHYZ. ZYOB ZYHZ by the corresponding angles theorem. Therefore,
YB. YO
by AA similarity. Similar triangles have proportional sides, therefore, X = X = Y
BZ OH
Yo = 1; YO = OH. O is the midpoint of YH by the definition of a midpoint. AO is the midsegment of AXYH and AO || XH by the
definition of midsegment. This establishes
YO
as a parallelogram. Using properties of a parallelogram, OH bisects XZ. By the
and C is the midpoint of XZ. YC is a median and meets the other two
definition of a bisector,
medians at O.
