A friend and I took a test yesterday for our geometry and discrete math course. The question that puzzled us is:

in triangle ABC, BC is produced to E and CB is produced to D. AC = AB and the ratio of sides is: AB : DB as CE:AB. (or AB/DB = CE/AB). PRove triangle ABD is similar to ACE.

Now our class basically all wrote they were similar and congruent by somehow proving that DB = CE. my friend and i wrote the same because thats what the teacher was expecting.

You can provethey are similar through SAS proof, IF you assume DB = CE. But how do you prove they are the same? We think you cant, andtherefore that ratio given does not work for all values that you can input for AB, AC, DB and CE. for example;

if AB = AC and AB = 6, and DB = 4. When you plug in these values into the ratio and isolate for CE, you get CE is 9. therefore CE does not equal DB.

our question is are we right? or we missing something really obivous here.

the answer the teacher said after the test was that the two triangles were a special case of similarity -> congruent. thank you in advance