User:Rayyanahmed1423/Rayyan Theorem

THE PROOF IS ORIGINAL AND DONE BY RAYYAN AHMED OR WIKIPEDIA USERNAME - Rayyanahmed1423

THE PROOF IS INDEPENDENT OF ANY OTHER MAJOR THEOREM RELATED TO THE SUBJECT.

THE  THEOREM : IF TWO LINES INTERSECT TWO PARALLEL LINES THEN THE INTERCEPTS  ARE IN THE SAME RATIO

LETS  DISCUSS  MY PROOF

LETS SEE THE DIAGRAM FIRST

GIVEN ELEMENTS :-

$$\angle A = \angle D

$$

$$\angle B = \angle E$$

$$\angle C = \angle F$$

$$OC = R  ,     PF = r   $$

PROOF :-

SINCE, IF ARCS SUBTEND EQUAL ANGLES AT THE CENTRE OR CIRCUMFERENCE THEN THE ARCS ARE SIMILAR

$$\left ( \frac{arcAC}{arcDF} \right )= \left ( \frac{\left ( \frac{2\angle B}{360 ^\circ} \right ) \times2\pi R}{\left ( \frac{2\angle E }{360 ^\circ} \right ) \times 2\pi r} \right ) = \left ( \frac{R}{r} \right ) $$ --- equation 1

$$\left ( \frac{arcAB}{arcDE} \right )= \left ( \frac{\left ( \frac{2\angle C}{360 ^\circ} \right ) \times2\pi R}{\left ( \frac{2\angle F }{360 ^\circ} \right ) \times 2\pi r} \right ) = \left ( \frac{R}{r} \right )   $$--- equation 2

$$\left ( \frac{arcBC}{arcEF} \right )= \left ( \frac{\left ( \frac{2\angle A}{360 ^\circ} \right ) \times2\pi R}{\left ( \frac{2\angle D }{360 ^\circ} \right ) \times 2\pi r} \right ) = \left ( \frac{R}{r} \right )   $$--- equation 3

AS FROM ABOVE EQUATIONS (1,2,3), WE CAN CONCLUDE FROM  EUCLID'S POSTULATES THAT "IF THINGS ARE EQUAL TO THE SAME OBJECT THEN THE OBJECTS SRE CONSIDERED TO BE EQUAL TO EACH OTHER "

FROM ABOVE :-

$$\left ( \frac{arcAB}{arcDE} \right ) = \left ( \frac{R}{r} \right ) $$

$$\left ( \frac{arcBC}{arcEF} \right ) = \left ( \frac{R}{r} \right ) $$

$$\left ( \frac{arcAC}{arcDF} \right ) = \left ( \frac{R}{r} \right ) $$

HENCE :-    $$\left ( \frac{arcAC}{arcDF} \right ) =  $$$$\left ( \frac{arcAB}{arcDE} \right ) =  $$$$\left ( \frac{arcBC}{arcEF} \right )   $$

IF ALL CHORDS ARE SIMILAR THEN THE CORRESPONDING ARCS ARE ALSO SIMILAR

Therefore:- $$\left ( \frac{AB}{DE} \right ) = \left ( \frac{AC}{DF} \right ) = \left ( \frac{BC}{EF} \right ) $$

THE PROOF IS ORIGINAL AND DONE BY RAYYAN AHMED OR WIKIPEDIA USERNAME - Rayyanahmed1423