BIGtheme.net http://bigtheme.net/ecommerce/opencart OpenCart Templates
Friday , July 28 2017
Home / Geometry / Prove that the middle points of equal chords of a circle are concyclic.

# Prove that the middle points of equal chords of a circle are concyclic.

General enunciation: We have to show that the middle points of equal chords of a circle are concyclic.

Particular enunciation: Consider, O is the centre of the circle ABCD, AB, CD and EF are three equal chords of it. M, N and P are the middle points of AB, CD and EF respectively. Let us prove that M, N and P are concyclic.

Construction: Let us join (O, M), (O, N) and (O, P).

Proof: Since M is the middle point of AB; OM is the joining line of the centre and the middle point of the chord AB, then OM ⊥ AB.

Similarly, OP⊥CD and ON ⊥ EF.

Now we know that equal chords of the circle equidistant from the circle.

∴ OM = ON = OP

Hence, if we draw a circle with the centre at O and OM or ON or as radius, it pass through the points M, N and P.

M, N and P are concyclic. (Proved)

Problem: Show that the equal chords drawn from two ends of the diameter on its opposite sides are parallel.

General enunciation: We have to prove that the equal chords drawn from two ends of the diameter on its opposite sides are parallel.

Particular Enunciation: Let, O is the centre of the circle ABCD and Ac is diameter. AB and CD are two equal chords are situated opposite sides of AC. We have to show that AB || CD.

Construction: (O, B) and (O, D) are joined.

Proof: In OAB and OCD we get,

OA = OC [radii of the same circle]

OB = OD [radii of the same circle]

and AB = CD [according to question]

∴ ∆ OAB ≅ ∆ OCD

∴ ∠ OAB = ∠OCD

i.e., ∠ BAC = ∠ACD, but these two are alternate angles and lie in the opposite sides of AC.

Hence, AC is secant of AB and CD.

∴ AB || CD

Problem: Show that the two parallel chords of a circle drawn from two ends of a diameter on its opposite sides are equal.

General Enunciation: We have to show that the two parallel chords of a circle drawn from two ends of a diameter on its opposite sides are equal.

Particular Enunciation: Let, O is the centre of the circle ABCD and AC is diameter. AB and CD are two parallel chords situated opposite sides of AC. Let us show that AB = CD.

Construction: Let us draw two perpendiculars OM and ON to the chords AB and CD respectively.

Proof: Since O is the centre of the circle and OM ⊥ AB, then we get, AM = ½ AB. Similarly, O is the centre of the circle and ON ⊥ CD, then we get, CN = ½ CD.

Now in right angles ∆ OAM and ∆ OCN we get,

∠OMA = ∠ ONC

∠OAM = ∠ OCN

And OA = OC

∴ ∆ OAM ≅ ∆ OCN

Then we get, AM = CN

∴ ½ AB = ½ CD

i.e., AB = CD (Showed)

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...