Geometry Proofs

How to prove that: The line drawn from the centre of a circle perpendicular to a chord bisects the chord. (𝒍𝒊𝒏𝒆 𝒇𝒓𝒐𝒎 𝒄𝒆𝒏𝒕𝒓𝒆 ⊥ 𝒕𝒐 𝒄𝒉𝒐𝒓𝒅). [With the given image]

How to prove that: The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. (𝒍𝒊𝒏𝒆 𝒇𝒓𝒐𝒎 𝒄𝒆𝒏𝒕𝒓𝒆 𝒕𝒐 𝒎𝒊𝒅𝒑𝒕 𝒐𝒇 𝒄𝒉𝒐𝒓𝒅) [With the given image]

How to prove that: The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circumference (on the same side of the chord as the centre). (∠ 𝒂𝒕 𝒄𝒆𝒏𝒕𝒓𝒆 = 𝟐 × ∠ 𝒂𝒕 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆) [With the given image]

(When drawing remember to label each angle the extension creates)

How to prove that: The opposite angles of cyclic quadrilateral are supplementary. (𝒐𝒑𝒑 ∠ ′𝒔 𝒐𝒇 𝒄𝒚𝒄𝒍𝒊𝒄 𝒒𝒖𝒂𝒅) [With the given image]

How to prove that: The angle between the tangent to a circle and a chord drawn from the point of contact are equal to the angle in the alternate segment. (𝒕𝒂𝒏 𝒄𝒉𝒐𝒓𝒅 𝒕𝒉𝒆𝒐𝒓𝒆𝒎) [With the given image - prove 𝐵𝑃̂𝑇 = 𝐴̂)

How to prove: Proportional division theorem [With the given image]

CONSTRUCTION:

1. Draw altitudes h (base AD) and K (base AE)

2. Join BE and DC to create ΔBDE and ΔCED

PROOF:

1. Area ΔADE / Area ΔBDE = (¹/₂ . AD . h) / (¹/₂ . BD . h) = AD / BD

2. Area ΔADE / Area ΔCDE = (¹/₂ . AE . h) / (¹/₂ . CE . h) = AE / CE

3. Area ΔBDE = Area ΔCDE (Same base, same height, lying between ∥ lines)

4. Area ΔADE / Area ΔBDE = Area ΔADE / Area ΔCDE

5. AD / BD = AE / BC

How to prove that: 2 triangles are similar [With the given image]

CONSTRUCTION:

1. Label P, where AP = DE

2. Label Q, where AQ = DF

3. Join P and Q to form PQ

PROOF:

1. In ΔAPQ and ΔDEF
   1. ∠A = ∠D Given
   2. AP = DE Construction
   3. AQ = DF Construction
   ΔAPQ ≡ ΔDEF (s,∠,s)

2. ∠APQ = ∠E and ∠AQP = ∠F (ΔAPQ ≡ ΔDEF)
   ∠APQ = ∠B = ∠E (Given)
   PQ ∥ BC (Corresponding ∠s are =)

3. AP / AB = AQ / AC (ΔABC; PQ ∥ BC)
   But AP = DE and AQ = DF

4. DE / AB = DF / AC