# given: ∠ abc, rs is the perpendicular bisector of ab, rt is the perpendicular bisector of bcprove: ar ≅ rc

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given: ∠ abc, rs is the perpendicular bisector of ab, rt is the perpendicular bisector of bc

prove: ar ≅ rc

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Step 1

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.

Step 2

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

Step 3

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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