I am sure this has been asked SOMEWHERE but I can't find it anywhere.
Question is simple: Will R1 chose the EIGRP route to R4 over the OSPF route, even though the OSPF route is clearly the better route?
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You cannot compare metrics between different routing protocols.
To solve that problem, we have Administrative Distance, which gives you a relative trustworthiness of a particular routing protocol, as defined by the router vendor. In particular, Cisco defined the AD of EIGRP to be lower (more trustworthy) than that of OSPF.
It is possible to change the AD.
Without changing any defaults, the same routes as learned by EIGRP and OSPF will install the EIGRP routes in the routing table because the AD for EIGRP is lower.
When two routing protocols are configured in router routing towards same destination . Router is designed to chosee routing protocol with least administrative distance (AD) value .
In this scenario OSPF & EIGRP is configured . So router picks up EIGRP because it's AD value is 90 compared to OSPF AD value 110 .
Upon chossing EIGRP protocal .Further to choose best path metric is calculated among available paths .
A router uses an order in which the best route to the destination is chosen. The order is:
Let apply this order one by one to your scenario. Let us assume that R4 is advertising same route via both the routing protocols. Hence, prefix length is equal. The second step is checking the administrative distance(AD) which tells the router how "trustworthy" a protocol is. The administrative distance of EIGRP is 90 and OSPF is 110 (in Cisco), hence EIGRP is preferred (Lower AD). The metric will never be compared since the AD already decided which route should be preferred. You can manually configure administrative distance value to help the situation.
router eigrp 100 distance eigrp 120 170
(First value is for internal EIGRP and second is for external EIGRP)