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In the given figure, from a point O in the interior of triangle ABC, perpendiculars OD, OE, and OF are drawn to the sides BC, CA, and AB respectively. Prove that:
(a) AF2+BD2+CE2=OA2+OB2+OC2OD2OE2OF2
(b) AF2+BD2+CE2=AE2+CD2+BF2
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Answer
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Hint: Here, we need to prove the given equations. We will use linear pair angles property, and Pythagoras’s theorem in three right angles to get three equations. Then, we will use the three equations to prove the required equation. Similarly, we will prove the equation in part (b) using the linear pair property, Pythagoras’s theorem, and the equation proved in part (a).

Complete step-by-step answer:
(a)
We will use the Pythagoras’s theorem to prove that AF2+BD2+CE2=OA2+OB2+OC2OD2OE2OF2.
From the figure, we can observe that the triangles AOF, BOD, and AOE are right angled triangles.
In the right angled triangle AOF, AO is the hypotenuse, AF is the base, and OF is the perpendicular.
Using the Pythagoras’s theorem in right angled triangle AOF, we get
AO2=AF2+OF2
Subtracting OF2 from both sides of the equation, we get
AF2=AO2OF2
In the right angled triangle BOD, BO is the hypotenuse, BD is the base, and OD is the perpendicular.
Using the Pythagoras’s theorem in right angled triangle BOD, we get
BO2=BD2+OD2
Subtracting OD2 from both sides of the equation, we get
BD2=BO2OD2
From the figure, we can observe that the angles CEO and AEO form a linear pair.
Therefore, we get
CEO+AEO=180
Substituting AEO=90 in the equation, we get
CEO+90=180
Subtracting 90 from both sides of the equation, we get
CEO=90
Therefore, the triangle COE is a right angled triangle.
In the right angled triangle COE, CO is the hypotenuse, CE is the base, and OE is the perpendicular.
Using the Pythagoras’s theorem in right angled triangle COE, we get
CO2=CE2+OE2
Subtracting OE2 from both sides of the equation, we get
CE2=CO2OE2
Now, adding both sides of the equations AF2=AO2OF2, BD2=BO2OD2, and CE2=CO2OE2, we get
AF2+BD2+CE2=AO2OF2+BO2OD2+CO2OE2
Rewriting the equation by rearranging the terms, we get
AF2+BD2+CE2=OA2+OB2+OC2OD2OE2OF2
Hence, we have proved that AF2+BD2+CE2=OA2+OB2+OC2OD2OE2OF2.
 (b)
From the figure, we can observe that the angles AFO and BFO form a linear pair.
Therefore, we get
AFO+BFO=180
Substituting AFO=90 in the equation, we get
BFO+90=180
Subtracting 90 from both sides of the equation, we get
BFO=90
Therefore, the triangle BOF is a right angled triangle.
From the figure, we can observe that the angles BDO and CDO form a linear pair.
Therefore, we get
BDO+CDO=180
Substituting BDO=90 in the equation, we get
90+CDO=180
Subtracting 90 from both sides of the equation, we get
CDO=90
Therefore, the triangle COD is a right angled triangle.
In the right angled triangle AOE, AO is the hypotenuse, AE is the base, and OE is the perpendicular.
Using the Pythagoras’s theorem in right angled triangle AOE, we get
AO2=AE2+OE2
Subtracting OE2 from both sides of the equation, we get
AE2=AO2OE2
In the right angled triangle BOF, BO is the hypotenuse, BF is the base, and OF is the perpendicular.
Using the Pythagoras’s theorem in right angled triangle BOF, we get
BO2=BF2+OF2
Subtracting OF2 from both sides of the equation, we get
BF2=BO2OF2
In the right angled triangle COD, CO is the hypotenuse, CD is the base, and OD is the perpendicular.
Using the Pythagoras’s theorem in right angled triangle COD, we get
CO2=CD2+OD2
Subtracting OD2 from both sides of the equation, we get
CD2=CO2OD2
We have proved that AF2+BD2+CE2=OA2+OB2+OC2OD2OE2OF2.
Grouping the terms using parentheses, we get
AF2+BD2+CE2=(OA2OE2)+(OB2OF2)+(OC2OD2)
Rewriting the terms of the expression, we get
AF2+BD2+CE2=(AO2OE2)+(BO2OF2)+(CO2OD2)
Substituting AO2OE2=AE2, BO2OF2=BF2, and CO2OD2=CD2 in the equation, we get
AF2+BD2+CE2=AE2+BF2+CD2
Hence, we have proved that AF2+BD2+CE2=AE2+BF2+CD2.

Note: We used the sum of two angles in a linear pair in the solution. The sum of all the angles lying on a line is equal to 180. These angles are said to form a linear pair.
We used the Pythagoras’s theorem in the solution to solve the solution. The Pythagoras’s theorem states that the square of the hypotenuse of a right angled triangle is equal to the sum of squares of the other two sides, that is Hypotenuse2=Perpendicular2+Base2.