given: ∠ abc, rs is the perpendicular bisector of ab, rt is the perpendicular bisector of bc
prove: ar ≅ rc
Please look at the diagram on the white board.
It should not take you long to realize that R is the point of intersection of two perpendicular bisectors. Hence, it must be the circumcentre of the triangle. A circumcentre is equidistant from all the three vertices of the triangle. Hence, AR = RC.
If you are looking for a more rigorous proof, please see it below,
Join the points B & R.
Consider the triangle ASR and BSR,
SR is common,
Angle ASR = Angle BSR = 90 degrees = right angle (RS is perpendicual bisector of AB)
AS = BS (The perpendicular bisector RS will bisect the AB at S and hence AS = BS)
Hence, by RHS rule of congruence, Triangle ASR is congruent to Trianlge BSR
Hence, AR = BR
Consider the triangle BTR and CTR,
TR is common,
Angle BTR = Angle CTR = 90 degrees = right angle (RT is perpendicual bisector of BC)
BT = CT (The perpendicular bisector R...
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