Question

sᴛᴀᴛᴇ ᴛʜᴀʟᴇs ᴛʜᴇᴏʀᴇᴍ...​

Answer 1

Explanation:

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements.

Proof on Thales theorem :

If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

Answer 2

$\huge\mathfrak{Answer:}$

➡️If a line is drawn parallel to the one side of the triangle then the other two sides gets divided in same ratio.

✳PROOF -

Construction: ABC is a triangle, DE is a line parallel to BC and intersecting AB at D and AC at E, i.e. DE || BC.

➡️Join C to D and B to E. Draw EM ⊥ AB and DN ⊥ AC.

➡️We need to prove that AD/DB = AE/EC

Proof:

➡️Area of a triangle, ADE = ½ × AD × EM

Similarly,

Ar(BDE) = ½ × DB × EM

Ar(ADE) = ½ × AE × DN

Ar(DEC) = ½ × EC × DN

Hence,

Ar(ADE)/Ar(BDE) = ½ × AD × EM / ½ × DB × EM = AD/DB

Similarly,

Ar(ADE)/Ar(DEC) = AE/EC

➡️Triangles DEC and BDE are on the same base, i.e. DE and between same parallels DE and BC.

Hence,

Ar(BDE) = Ar(DEC)

➡️From the above equations, we can say that

AD/DB = AE/EC.

➡️Hence, proved.