In פרשת בלק when בלעם wanted to curse the jews the פסוק says
"ןיפתח 'ה את פי האתון ותאמר לבלעם מה עשיתי לך כי הכיתני זה שלש רגלים
"And Hashem opened the mouth of the donkey: What did i do that caused you too hit me these three times?
The question that needs to be asked is: why did the פסוק use the word רגלים
as opposed to the more commonly used word פעמים?
רש"י answers רמז לו: אתה מבקש לעקור אמה החוגגת שלש רגלים בשנה Hashem is telling us that בלעם won't be able to destroy the Jews because the זכות of the שלש רגלים is too great. בלעם however, would not give up that easily.
"וישת אל המדבר פניו"
בלעם looked towards the מדבר... to invoke the the anger oh 'ה by bringing up the חטא העגל. At the חטא העגל itself, משה davens and mentions the אבות because he wants 'ה to have mercy on בני ישראל and the זכות אבות protects us from destruction. yet, what happened to the שבועה that 'ה promised to destroy בני ישרשל?
There is a story that a king once
decreed that his son should be stoned by a big boulder. On the day of the stoning the king asked to rescind his decree but was unable to do so. So he gave a new decree saying that a big boulder would be cut into small pebbles, with which the the prince would be stoned. Yes the prince would be hurt but no, he would not be killed So too by חטא העגל בני ישראל were not destroyed, but in any generation that חס ושלום, בני ישראל would do an עבירה they are punished with the "pebbles" of חטא העגל.
Our protection from this suffering comes from the שלש רגלים, as it says by the עגל
כי בשש משה.
רש"י explains that משה told בני ישראל that he would be back on the 40th day in the sixth hour. Along came the שטן and darkened the sky. בני ישראל said that surely the sixth hour has passed and משה has not yet returned. They then made the עגל.
בני ישראל sinned because they miscalculated the times. Using the כלל of בטל בשישים we can cancel these 6 hours, with the total of 360 hours, which is equal to 15 days. These 15 days are 7 days of פסח, the 7 days of סכות and one day of שבעות.
Therefore by keeping all the days of the רגלים as hinted to by the אתון we counteract the punishment of the חטא העגל.
What about the כהנים? They weren't involved in the חטא.
Because the כהנים were not involved, they have the job to bless the rest of כלל ישראל on these 15 days. The 3 פסוקים of ברכת כהנים correspond to the 3 אבות, the 15 words correspond to the 15 days of the רגלים and the 60 letter correspond to the rule of בטל בשישים. We can see from here that the שלש רגלים have the ability to counteract the punishment of the חטא העגל and that we should keep them fully.