 Hello and welcome to the session. I am Deepika here. Let's discuss a question which says if a line is drawn parallel to one side of a triangle to intersect the other two sides in Distinct points prove that the other two sides are divided in the same ratio use the upper proof the following In given figure DE is parallel to BC and BD is equal to CE. Prove that triangle ABC is an isocialized triangle. So let's start the solution. Now we are given a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points. So we are given a triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively. So we can write given a triangle ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively. Now we have to prove that the other two sides are divided in the same ratio. That is we have to prove AD over DB is equal to AE over EC. Now let us join VE and CG and then draw a perpendicular from D on AC and a perpendicular from E on AB. So we can write join VE and CG and then draw DBM perpendicular on AC and EN perpendicular AB. Now area of triangle ADE is given by 1 over 2 into base into height that is area of triangle ADE is equal to 1 over 2 into base AD into height EN. Similarly, area of triangle BDE is equal to 1 over 2 into base DB into height EN. Again, area of triangle ADE can be written as 1 over 2 into base AE into height DM and area of triangle DEC is equal to 1 over 2 into base EC into height DM. Therefore, area of triangle ADE over area of triangle BDE is equal to now area of triangle ADE is equal to 1 over 2 into AD into EN and area of triangle BDE is equal to 1 over 2 into DB into EN. Therefore, area of triangle ADE upon area of triangle BDE is equal to 1 over 2 into AD into EN over 1 over 2 DB into EN and this is further equal to AD over DB. Let us give this as number 1 that is we have area of triangle ADE upon area of triangle BDE is equal to AD over DB. Again, area of triangle ADE upon area of triangle DEC is equal to now area of triangle ADE is equal to half AE into DM and area of triangle DEC is equal to half EC into DM. So, area of triangle ADE upon area of triangle DEC is equal to half AE into DM over half EC into DM and this is further equal to AE over EC. Let us give this as number 2. Now, here we observe that triangle BDE and triangle DEC are on the same base DE and between the same parallels BC and DE. So, area of triangle BDE is equal to area of triangle DEC. So, we can write since triangle BDE and DEC are on the same base DE and between the same parallels BC and DE. So, area of triangle BDE is equal to area of triangle DEC. Let us give this as number 3. Now, from 1 we have area of triangle ADE upon area of triangle BDE is equal to AD over DB and from 2 we have area of triangle ADE upon area of triangle DEC is equal to AE upon EC and from 3 we have area of triangle BDE is equal to area of triangle DEC. Therefore, from 1, 2 and 3 we have AD over DB is equal to AE over EC. Hence, we have proved that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points the other two sides are divided in the same ratio. Now, by using this result, again we have to prove in given figure DE is parallel to BC and BDE is equal to CE. We have to prove that triangle ABC is an isocialist triangle. So, in given figure we have DE is parallel to BC and BDE is equal to CE. So, therefore, by using above result we have AD over DB is equal to AE over EC but we are given BDE is equal to CE. Therefore, AD over DB is equal to AE over DB or we can say AD is equal to AE. Let us give this as number 4 and this as number 5. So, on adding 4 and 5 we get BDE plus AD is equal to CE plus AE or AB is equal to AC. Now, we know that if any two sides of a triangle are equal then it is an isocialist triangle. So, in triangle ABC we have AB is equal to AC. Hence, triangle ABC is an isocialist triangle. Hence, we had proved the above question. This completes our session. I hope the solution is clear to you. Bye and have a nice day.