Question

25*. In figure 3.103, seg AD I side BC,
seg BE 1 side AC, seg CF I side
AB. Ponit O is the orthocentre. Prove
that , point 0 is the incentre of
A DEF
​

Answer 1

Explanation:

Given that seg AD ⊥ side BC,  seg BE ⊥ side AC, seg CF ⊥ side  AB.

Point O is the orthocentre.

To prove that the point O is the incentre of ΔDEF

Let us join the points DE, DF and EF

From the figure, we can see that

$\angle {AFO}+\angle {AEO}=90^{\circ}+90^{\circ}=180^{\circ}$

Since, we know that "if a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cylic".

Thus, the quadrilateral AEOF is cylic.

Since, the angles inscribed in the same arc are congruent.

Then, we have,

$\angle {OAE}=\angle {OFE}$ -----------(1)

$\angle {BFO}+\angle {BDO}=90^{\circ}+90^{\circ}=180^{\circ}$

Applying the property, "if a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cylic".

Then, the quadrilateral BFOD is cyclic.

Since, the angles inscribed in the same arc are congruent.

Then, we have,

$\angle {OBD}=\angle{OFD}$ ----------(2)

Let us consider the triangle BCE

By angle sum property, we have,

$\angle {BCE}+\angle {CBE}=90^{\circ}$ -----------(3)

Let us consider the triangle ACD

By angle sum property, we get,

$\angle {DAC}+\angle {ACD}=90^{\circ}$ ----------(4)

Equating the equations (3) and (4), we have,

$\angle {DAC}+\angle{ACD}=\angle {BCE}+\angle {CBE}$

               $\angle {DAC}=\angle {CBE}$  ----------(5)

From the equations (1), (2) and (5), we have,

$\angle{OFE}=\angle {OFD}$

Since OF is the bisector of $\angle E F D$

Therefore, the point O is the incentre of ΔDEF

Hence proved

Learn more:

(1) In the given fig.,BE perpendicular AC. AD is any line from A to BC intersecting BE at H. P, Q and R are mid-points of AH, AB and BC respectively, then prove that angle PQR=90o.

brainly.in/question/1134535

(2) O is any point inside a triangle abc.the bisectors of angle aob,angle boc and angle coa meet the sides ab,bc and ca in point d.e and f respectively.show that

1.)ad*be*cf=db*ec*fa

brainly.in/question/1302503

Answer 2

It is given that seg AD 1 side BC, seg BE 1 side AC and seg CF 1 side BC. O is the

orthocentre of AABC

Join DE, EF and DF.

LAFO + LAEO = 90° +90º = 180° Quadrilateral AEOF is cycle. (If a pair

of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic) - 2OAE = 20FE .(1) (Angles inscribed in the same arc are congruent) BF + <B00 - 90" + 90° - 180° Quadrilateral BFOD is cyclic. (If a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic) - 2 OBD - 2 OF D (2) (Angles inscribed in the same arc are cangruent) From (3) and (4), we get LDAC + LACD = LBCE + LCBE - LDAC = LCBE (5) From (1). (2) and (5), we get LIFE = COFD on bror

- OF is the bisector of ZEFD. Similarly, OE and OD are the bisectors of DEF and EDF, respectively, Hence. O is the incentre of ADEF, (Incentre of a triangle is the point of intersection of its angle bisectors)